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This is PART 8: Centers X(14001) - X(16000)

PART 1: Introduction and Centers X(1) - X(1000) PART 2: Centers X(1001) - X(3000) PART 3: Centers X(3001) - X(5000)
PART 4: Centers X(5001) - X(7000) PART 5: Centers X(7001) - X(10000) PART 6: Centers X(10001) - X(12000)
PART 7: Centers X(12001) - X(14000) PART 8: Centers X(14001) - X(16000) PART 9: Centers X(16001) - X(18000)
PART 10: Centers X(18001) - X(20000) PART 11: Centers X(20001) - X(22000) PART 12: Centers X(22001) - X(24000)
PART 13: Centers X(24001) - X(26000) PART 14: Centers X(26001) - X(28000) PART 15: Centers X(28001) - X(30000)
PART 16: Centers X(30001) - X(32000) PART 17: Centers X(32001) - X(34000) PART 18: Centers X(34001) - X(36000)
PART 19: Centers X(36001) - X(38000) PART 20: Centers X(38001) - X(40000) PART 21: Centers X(40001) - X(42000)
PART 22: Centers X(42001) - X(44000) PART 23: Centers X(44001) - X(46000) PART 24: Centers X(46001) - X(48000)
PART 25: Centers X(48001) - X(50000) PART 26: Centers X(50001) - X(52000) PART 27: Centers X(52001) - X(54000)
PART 28: Centers X(54001) - X(56000) PART 29: Centers X(56001) - X(58000) PART 30: Centers X(58001) - X(60000)
PART 31: Centers X(60001) - X(62000) PART 32: Centers X(62001) - X(64000) PART 33: Centers X(64001) - X(66000)
PART 34: Centers X(66001) - X(68000) PART 35: Centers X(68001) - X(70000) PART 36: Centers X(70001) - X(72000)

X(14001) =  EULER LINE INTERCEPT OF X(6)X(3926)

Barycentrics    3a4 + (b2 + c2)2 : :

X(14001) lies on these lines: {2, 3}, {6, 3926}, {32, 69}, {39, 3618}, {76, 7735}, {83, 7736}, {99, 7738}, {141, 3053}, {148, 7932}, {187, 3619}, {193, 3933}, {194, 10336}, {315, 3972}, {316, 7930}, {538, 5319}, {543, 7902}, {574, 7889}, {590, 5491}, {615, 5490}, {620, 7808}, {626, 7737}, {754, 7869}, {1007, 2548}, {1056, 6645}, {1058, 4366}, {1249, 8863}, {1285, 3314}, {1352, 13335}, {1384, 3620}, {1975, 5286}, {1992, 5007}, {2549, 7816}, {3329, 7891}, {3407, 7793}, {3589, 5013}, {3734, 3767}, {3763, 5023}, {5008, 7855}, {5206, 6292}, {5304, 7754}, {5305, 6392}, {5475, 7874}, {6390, 9605}, {6781, 7935}, {7612, 7697}, {7739, 7781}, {7745, 7778}, {7747, 7867}, {7748, 7852}, {7750, 7868}, {7753, 7888}, {7756, 7913}, {7759, 7880}, {7761, 7915}, {7762, 7881}, {7772, 7863}, {7774, 7787}, {7782, 7859}, {7783, 7875}, {7785, 7945}, {7796, 12150}, {7799, 7878}, {7802, 7944}, {7812, 7909}, {7823, 7931}, {7828, 11185}, {7830, 7914}, {7838, 7908}, {7847, 7943}, {7858, 7870}, {7921, 7947}, {8716, 9607}, {10350, 10788}, {10352, 10359}

X(14001) = reflection of X(33223) in X(2)
X(14001) = complement of X(32974)
X(14001) = anticomplement of X(7866)
X(14001) = crossdifference of every pair of points on line {647, 2514}
X(14001) = orthocentroidal-circle-inverse of X(14064)
X(14001) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2,3,16043), (2,4,14064), (2,5,32969), (2,20,6656) (2, 384, 4), (2, 3091, 7887), (2, 3523, 11285), (2, 3552, 7791), (2, 6658, 7933), (2, 7907, 3525), (3, 7819, 2), (6, 7789, 3926), (32,7794,14023), (32, 7795, 69), (32, 7820, 7795), (83, 7763, 7736), (83, 7835, 7763), (99, 7803, 7738), (99, 7846, 7803), (141, 3053, 3785), (187, 7822, 7800), (384,5025,14035), (384, 7892, 2), (384, 7901, 11361), (550, 8364, 11287), (1003, 6656, 20), (1975, 7792, 5286), (2548, 3788, 1007), (3552, 7791, 376), (3618, 6337, 39), (3734, 6680, 3767), (3788, 7804, 2548), (3972, 7832, 315), (5007, 7758, 1992), (5007, 7801, 7758), (6661, 7807, 7770), (7770, 7807, 2), (7781, 7829, 7739), (7787, 7836, 7774), (7800, 7822, 3619), (7816, 7834, 2549), (7819, 8369, 3), (7887, 8370, 3091), (8365, 11318, 2), (8366, 8370, 2), (8368, 11286, 2), (11291, 11292, 4), (37172,37173,376), (37340,37341,381)

X(14002) =  EULER LINE INTERCEPT OF X(32)X(111)

Barycentrics    a^2 (2a^4 - 2 b^4 + 5 b^2 c^2 - 2 c^4) : :

X(14002) lies on these lines: {2, 3}, {32, 111}, {51, 9544}, {110, 576}, {182, 5643}, {323, 11477}, {353, 5038}, {511, 10546}, {568, 5609}, {575, 1495}, {669, 5466}, {1078, 6031}, {1302, 13530}, {1975, 5971}, {1992, 2930}, {2076, 8617}, {2502, 13330}, {2936, 8596}, {3053, 11580}, {3060, 3292}, {3066, 6800}, {3284, 10985}, {3622, 8185}, {3623, 11365}, {5007, 9465}, {5206, 8585}, {5265, 9658}, {5281, 9673}, {6236, 9084}, {6593, 9971}, {7664, 7752}, {7665, 7785}, {7747, 10418}, {7754, 9870}, {7766, 9149}, {9545, 9781}, {9590, 9779}, {9703, 13451}

X(14002) = circumcircle-inverse of X(37907)

X(14003) =  EULER LINE INTERCEPT OF X(32)X(125)

Barycentrics    a^8 - 2 a^6 b^2 + b^8 - 2 a^6 c^2 - 2 b^4 c^4 + c^8 : :

X(14003) lies on these lines: {2, 3}, {32, 125}, {247, 5106}, {571, 6697}, {5650, 7822}

X(14004) =  EULER LINE INTERCEPT OF X(33)X(92)

Barycentrics    (a^2 - ab - ac - bc)(a^2 + b^2 - c^2)(a^2 - b^2 + c^2) : :

X(14004) lies on these lines: {1, 5342}, {2, 3}, {33, 92}, {34, 4666}, {200, 318}, {242, 1824}, {281, 6605}, {1430, 11019}, {1785, 1860}, {1826, 2201}, {1827, 10025}, {1843, 2905}, {1862, 7140}, {1887, 1940}, {1890, 5263}, {3332, 11433}, {3957, 6198}, {3996, 4043}, {4847, 5081}, {7282, 9436}, {7952, 10578}

X(14005) =  EULER LINE INTERCEPT OF X(8)X(86)

Barycentrics    (a + b)(a + c)(a^2 + ab + 2b^2 + ac + 4bc + 2c^2) : :

X(14005) lies on these lines: {1, 4720}, {2, 3}, {8, 86}, {10, 81}, {12, 5323}, {58, 750}, {274, 1390}, {333, 9780}, {388, 1014}, {899, 4281}, {1043, 3616}, {1213, 1778}, {1255, 2901}, {1412, 9578}, {2287, 5750}, {3306, 10461}, {3617, 8025}, {3624, 4653}, {3633, 4803}, {3679, 4658}, {3739, 5262}, {3786, 3868}, {4267, 4413}, {4278, 5251}, {4877, 9579}, {10455, 12435}

X(14006) =  EULER LINE INTERCEPT OF X(31)X(92)

Barycentrics    (a + b)(a + c)(a - b - c)(a^2 + bc)(a^2 + b^2 - c^2)(a^2 -b^2 + c^2) : :

X(14006) lies on these lines: {2, 3}, {8, 2907}, {31, 92}, {33, 1808}, {270, 318}, {894, 3955}, {987, 1896}, {1172, 7155}, {2268, 2326}, {2328, 3923}

X(14006) = cevapoint of X(171) and X(7009)
X(14006) = X(i)-isoconjugate of X(j) for these (i,j): {65, 7015}, {71, 1432}, {72, 1431}, {73, 256}, {201, 1178}, {226, 7116}, {228, 7249}, {257, 1409}, {307, 904}, {893, 1214}, {1231, 7104}, {1402, 7019}, {1410, 4451}
X(14006) = {X(2074),X(5136)}-harmonic conjugate of X(1982)
X(14006) = barycentric product X(i)*X(j) for these {i,j}: {27, 7081}, {29, 894}, {33, 8033}, {261, 1840}, {270, 3963}, {286, 2329}, {314, 7119}, {333, 7009}, {648, 3907}, {811, 3287}, {1172, 1909}, {1237, 2189}, {1920, 2299}, {2322, 7176}, {2332, 7205}, {4183, 7196}
X(14006) = barycentric quotient X(i)/X(j) for these {i,j}: {27, 7249}, {28, 1432}, {29, 257}, {171, 1214}, {172, 73}, {284, 7015}, {333, 7019}, {894, 307}, {1172, 256}, {1474, 1431}, {1840, 12}, {1909, 1231}, {2189, 1178}, {2194, 7116}, {2204, 904}, {2295, 201}, {2299, 893}, {2322, 4451}, {2329, 72}, {2330, 71}, {3287, 656}, {3907, 525}, {4032, 6356}, {4095, 3695}, {4140, 4064}, {4459, 4466}, {4477, 8611}, {7009, 226}, {7081, 306}, {7119, 65}, {7122, 1409}, {7175, 1439}, {8033, 7182}

X(14007) =  EULER LINE INTERCEPT OF X(10)X(86)

Barycentrics    (a + b)(a + c)(a^2 + 2ab + 3b^2 + 2ac + 6bc + 3c^2) : :

X(14007) lies on these lines: {2, 3}, {8, 5333}, {10, 86}, {58, 3634}, {81, 9780}, {274, 4385}, {333, 1698}, {942, 3786}, {1014, 5261}, {1043, 1125}, {1213, 1330}, {1434, 5290}, {1834, 6707}, {3216, 5331}, {3622, 4720}, {3635, 4803}, {5208, 5439}, {5323, 10588}, {5437, 10461}

X(14008) =  EULER LINE INTERCEPT OF X(11)X(81)

Barycentrics    (a + b)(a + c)(b^2 + ac - c^2)(ab - b^2 + c^2) : :

X(14008) lies on these lines: {2, 3}, {11, 81}, {42, 80}, {58, 7741}, {333, 11680}, {1043, 11681}, {1437, 3615}, {1746, 5012}, {2886, 5235}, {3060, 10478}, {3583, 4276}, {3720, 5443}, {3741, 11813}, {3816, 5333}, {3829, 4921}, {4267, 10896}, {4653, 7951}

X(14009) =  EULER LINE INTERCEPT OF X(11)X(86)

Barycentrics    (a + b)(a + c)(ab^3 - b^4 + a^2 bc + 2b^2 c^2 + ac^3 - c^4) : :

X(14009) lies on these lines: {2, 3}, {11, 86}, {12, 1043}, {42, 5086}, {81, 11269}, {226, 5208}, {256, 3120}, {310, 7249}, {333, 2651}, {908, 3786}, {1834, 5331}, {3485, 10453}, {3741, 12047}, {3822, 4653}, {4892, 10129}, {9612, 10461}, {10455, 10886}

X(14010) =  EULER LINE INTERCEPT OF X(11)X(124)

Barycentrics    (a + b)(a + c)(a - b - c)^2 (b - c)^2 (a^2 b - b^3 + a^2 c - 2abc + b^2 c + bc^2 - c^3) : :

X(14010) lies on these lines: {2, 3}, {11, 124}

X(14010) = complement of X(7451)

X(14011) =  EULER LINE INTERCEPT OF X(12)X(86)

Barycentrics    (a + b)(a + c)(a - b - c)(-ab^3 - b^4 + a^2 bc + 2b^2 c^2 - ac^3 - c^4) : :

X(14011) lies on these lines: {2, 3}, {11, 1043}, {12, 86}, {58, 3814}, {81, 11681}, {333, 1329}, {1210, 5208}, {2303, 9596}, {3193, 5230}, {3701, 4518}, {3786, 6734}, {3794, 3831}, {3825, 4653}, {10455, 10887}

X(14012) =  EULER LINE INTERCEPT OF X(31)X(75)

Barycentrics    (a + b)(a + c)(a^4 + a^2 b^2 + a^2 bc + b^3 c + a^2 c^2 + 2b^2 c^2 + bc^3) : :

X(14012) lies on these lines: {2, 3}, {31, 75}, {58, 4362}, {81, 3891}, {1215, 5009}, {2363, 10457}, {5311, 10458}

X(14013) =  EULER LINE INTERCEPT OF X(19)X(86)

Barycentrics    (a + b)(a + c)(a^2+b^2-c^2)(a^2 + 2ab - b^2 + 2ac - c^2)(a^2 - b^2 + c^2) : :

X(14013) lies on these lines: {2, 3}, {19, 86}, {81, 607}, {92, 274}, {332, 1958}, {333, 7119}, {653, 1880}, {662, 1474}, {1434, 5236}

X(14014) =  EULER LINE INTERCEPT OF X(19)X(81)

Barycentrics    a(a + b)(a + c)(a^2 + b^2 - c^2)(a^2 - b^2 + c^2)(a^2 + 2ab + b^2 + 2ac + 4bc + c^2) : :

X(14014) lies on these lines: {2, 3}, {19, 81}, {278, 1014}, {1171, 1396}, {1848, 5333}

X(14015) =  EULER LINE INTERCEPT OF X(19)X(112)

Barycentrics    a(a + b)(a + c)(a^2 + b^2 - c^2)(a^2 - b^2 + ^2)(a^3 + b^3 + abc + b^2 c + bc^2 + c^3) : :

X(14015) lies on these lines: {2, 3}, {19, 112}, {935, 12030}, {1474, 4653}, {2906, 11396}

X(14016) =  EULER LINE INTERCEPT OF X(19)X(91)

Barycentrics    (a + b)(a + c)(a^2 + b^2 - c^2)(a^2 - b^2 + c^2)(a^4 - 2a^2 b^2 + b^4 - 2a^2 bc - 2 a^2 c^2 - 2 b^2 c^2 + c^4) : :

X(14016) lies on these lines: {2, 3}, {19, 91}, {58, 225}, {81, 1068}, {1780, 1838}, {1826, 4877}, {1869, 10572}, {2360, 12047}, {5307, 12514}

X(14017) =  EULER LINE INTERCEPT OF X(19)X(35)

Barycentrics    a^2 (a^2 + b^2 - c^2)(a^2 - b^2 + c^2)(a^4 - 2a^2 b^2 + b^4 - 2a^2 bc - 2ab^2 c - 2a^2 c^2 - 2abc^2 + c^4) : :

X(14017) lies on these lines: {2, 3}, {6, 1175}, {19, 35}, {56, 3418}, {243, 1324}, {278, 1612}, {386, 2299}, {579, 1474}, {942, 11363}, {993, 1891}, {1058, 10835}, {1437, 5751}, {1602, 7742}, {1843, 5138}, {1848, 5248}, {1974, 4260}, {2204, 8743}, {2354, 7295}, {3085, 10830}, {3485, 11365}, {3486, 9798}, {3601, 7713}, {5090, 5791}, {5146, 5172}, {5285, 12514}, {6197, 11248}, {8185, 10572}

X(14018) =  EULER LINE INTERCEPT OF X(19)X(46)

Barycentrics    (a^2+b^2-c^2)(a^2-b^2+c^2)(a^3+3 a^2 b+a b^2-b^3+3 a^2 c+4 a b c+b^2 c+a c^2+b c^2-c^3) : :

X(14018) lies on these lines: {1, 1869}, {2, 3}, {10, 5307}, {19, 46}, {34, 386}, {57, 225}, {65, 278}, {610, 5747}, {942, 1068}, {1118, 1454}, {1155, 5146}, {1698, 1826}, {1841, 4261}, {1848, 12609}, {1861, 10479}, {1867, 5791}, {1868, 5044}, {2182, 5746}, {2217, 3418}, {5132, 11398}, {7009, 10449}

X(14019) =  EULER LINE INTERCEPT OF X(1)X(120)

Barycentrics    a^4 b^2-b^6+4 a^4 b c+4 a b^4 c+a^4 c^2-8 a^2 b^2 c^2+b^4 c^2+4 a b c^4+b^2 c^4-c^6 : :

X(14019) lies on these lines: {1, 120}, {2, 3}

X(14020) =  EULER LINE INTERCEPT OF X(8)X(45)

Barycentrics    2 a^4- a^3 b-3 a^2 b^2-a b^3-b^4-a^3 c-6 a^2 b c-6 a b^2 c-b^3 c-3 a^2 c^2-6 a b c^2-a c^3-b c^3-c^4 : :

X(14020) lies on these lines: {2, 3}, {8, 45}, {51, 3877}, {1698, 9324}, {4653, 5741}, {5248, 6187}

X(14021) =  EULER LINE INTERCEPT OF X(7)X(37)

Barycentrics    a^5 + 3 a^4 b-2 a^3 b^2-2 a^2 b^3+a b^4-b^5+3 a^4 c-4 a^2 b^2 c+b^4 c-2 a^3 c^2-4 a^2 b c^2-2 a b^2 c^2-2 a^2 c^3+a c^4+b c^4-c^5 : :

X(14021) lies on these lines: {2, 3}, {7, 37}, {63, 3730}, {344, 5279}, {579, 5738}, {948, 3188}, {1104, 4313}, {1212, 5813}, {3008, 4304}, {3772, 9598}, {5736, 5746}, {8053, 11677}

X(14022) =  EULER LINE INTERCEPT OF X(9)X(11)

Barycentrics    (a - b - c)(a^3 b^2-a^2 b^3-a b^4+b^5-4 a^3 b c-a^2 b^2 c-3 b^4 c+a^3 c^2-a^2 b c^2+2 a b^2 c^2+2 b^3 c^2-a^2 c^3+2 b^2 c^3-a c^4-3 b c^4+c^5) : :

X(14022) lies on these lines: {2, 3}, {9, 11}, {12, 5436}, {72, 496}, {200, 1837}, {218, 11269}, {226, 3660}, {329, 5729}, {497, 1260}, {908, 5728}, {950, 1329}, {960, 3813}, {1728, 6763}, {1998, 5722}, {3419, 3820}, {3825, 12572}, {3874, 11019}, {3925, 4512}, {5328, 5809}, {8568, 8804}, {8580, 10826}, {10582, 11375}

X(14022) = complement of X(1004)

X(14023) =  X(20)X(538)∩X(32)X(69)

Barycentrics    3a4 - (b2 + c2)2 : :
X(14023) = 8 X(546) - 9 X(7615) = 10 X(3) - 9 X(7618) = 9 X(7618) - 5 X(7758) = 5 X(631) - 4 X(7764) = 3 X(2) - 4 X(7780) = 3 X(376) - 2 X(7781) = 8 X(3) - 9 X(8182) = 4 X(7758) - 9 X(8182) = 4 X(7618) - 5 X(8182) = 2 X(5) - 3 X(8667) = 4 X(548) - 3 X(8716) = 4 X(140) - 3 X(9766) = 11 X(7758) - 18 X(11165) = 11 X(7618) - 10 X(11165) = 11 X(3) - 9 X(11165) = 11 X(8182) - 8 X(11165) = 10 X(632) - 9 X(11184)

X(14023) lies on these lines: {2, 5007}, {3, 524}, {4, 754}, {5, 8667}, {6, 7767}, {20, 538}, {32, 69}, {39, 193}, {76, 7737}, {140, 9766}, {182, 6308}, {183, 2548}, {187, 439}, {230, 7776}, {315, 385}, {376, 7781}, {543, 3529}, {546, 7615}, {548, 8716}, {574, 7890}, {599, 7819}, {620, 7916}, {626, 7735}, {631, 7764}, {632, 11184} et al

X(14023) = reflection of X(i) in X(j) for these (i,j): (4, 7751), (6309, 5188), (7758,3), (7759,7780)
X(14023) = anticomplement of X(7759)
X(14023) = crossdifference of every pair of points on line {2514,8665}
X(14023) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6, 7767, 7800), (32, 69, 7795), (32,7794,14001), (32, 7826, 69), (76,20065,14023), (183, 7762, 2548), (187, 7855, 3926), (193, 3785, 39), (315, 385, 3767), (385, 7893, 315), (5007, 7854, 2), (6179, 7768, 2), (7759, 7780, 2)

X(14024) =  CEVAPOINT OF X(238) AND X(242)

Barycentrics    (a + b)(a + c) (a - b - c) (a^2 - bc) (a^2 + b^2 - c^2) (a^2 - b^2 + c^2) : :

X(14024) lies on these lines: {8, 29}, {27, 1851}, {28, 330}, {239, 242}, {318, 2212} et al

X(14024) = cevapoint of X(238) and X(242)
X(14024) = X(29)-Hirst inverse of X(1172)
X(14024) = X(i)-isoconjugate of X(j) for these (i,j): {65, 295}, {73, 291}, {201, 741}, {226, 2196}, {228, 7233}, {292, 1214}, {307, 1911}, {335, 1409}, {337, 1402}, {1231, 1922}, {1254, 1808}, {1410, 4518}, {1439, 7077}
X(14024) = barycentric product X(i)*X(j) for these {i,j}: {27, 3685}, {28, 3975}, {29, 239}, {242, 333}, {270, 3948}, {286, 3684}, {314, 2201}, {350, 1172}, {648, 3716}, {811, 4435}, {1447, 2322}, {1474, 4087}, {1874, 7058}, {1921, 2299}, {4183, 10030}, {5009, 7017}
X(14024) = barycentric quotient X(i)/X(j) for these {i,j}: {27, 7233}, {29, 335}, {238, 1214}, {239, 307}, {242, 226}, {284, 295}, {333, 337}, {350, 1231}, {419, 4032}, {862, 2171}, {1172, 291}, {1429, 1439}, {1874, 6354}, {1914, 73}, {2189, 741}, {2194, 2196}, {2201, 65}, {2204, 1911}, {2210, 1409}, {2238, 201}, {2299, 292}, {2322, 4518}, {2332, 7077}, {3684, 72}, {3685, 306}, {3716, 525}, {3747, 2197}, {3985, 3695}, {4124, 4466}, {4183, 4876}, {4433, 3949}, {4435, 656}, {5009, 222}, {7054, 1808}

X(14025) =  (name pending)

Barycentrics    a (a^12-6 a^10 b^2+15 a^8 b^4-20 a^6 b^6+15 a^4 b^8-6 a^2 b^10+b^12+4 a^10 b c+4 a^9 b^2 c-8 a^8 b^3 c-16 a^7 b^4 c+24 a^5 b^6 c+8 a^4 b^7 c-16 a^3 b^8 c-4 a^2 b^9 c+4 a b^10 c-6 a^10 c^2+4 a^9 b c^2+2 a^8 b^2 c^2+20 a^6 b^4 c^2-8 a^5 b^5 c^2-28 a^4 b^6 c^2+18 a^2 b^8 c^2+4 a b^9 c^2-6 b^10 c^2-8 a^8 b c^3+16 a^6 b^3 c^3-16 a^5 b^4 c^3-8 a^4 b^5 c^3+32 a^3 b^6 c^3-16 a b^8 c^3+15 a^8 c^4-16 a^7 b c^4+20 a^6 b^2 c^4-16 a^5 b^3 c^4+26 a^4 b^4 c^4-16 a^3 b^5 c^4-12 a^2 b^6 c^4-16 a b^7 c^4+15 b^8 c^4-8 a^5 b^2 c^5-8 a^4 b^3 c^5-16 a^3 b^4 c^5+8 a^2 b^5 c^5+24 a b^6 c^5-20 a^6 c^6+24 a^5 b c^6-28 a^4 b^2 c^6+32 a^3 b^3 c^6-12 a^2 b^4 c^6+24 a b^5 c^6-20 b^6 c^6+8 a^4 b c^7-16 a b^4 c^7+15 a^4 c^8-16 a^3 b c^8+18 a^2 b^2 c^8-16 a b^3 c^8+15 b^4 c^8-4 a^2 b c^9+4 a b^2 c^9-6 a^2 c^10+4 a b c^10-6 b^2 c^10+c^12) : :

See Le Viet An and Peter Moses, Hyacinthos 26377.

X(14025) lies on these lines: (pending)

X(14026) =  (name pending)

Barycentrics    a^6 b-4 a^5 b^2-2 a^4 b^3+7 a^3 b^4-3 a b^6+b^7+a^6 c-2 a^5 b c+11 a^4 b^2 c-5 a^3 b^3 c-11 a^2 b^4 c+7 a b^5 c-b^6 c-4 a^5 c^2+11 a^4 b c^2-18 a^3 b^2 c^2+13 a^2 b^3 c^2+3 a b^4 c^2-3 b^5 c^2-2 a^4 c^3-5 a^3 b c^3+13 a^2 b^2 c^3-14 a b^3 c^3+3 b^4 c^3+7 a^3 c^4-11 a^2 b c^4+3 a b^2 c^4+3 b^3 c^4+7 a b c^5-3 b^2 c^5-3 a c^6-b c^6+c^7 : :

See Le Viet An, Peter Moses, and César Lozada, Hyacinthos 26382 and Hyacinthos 26383.

X(14026) lies on these lines: {3, 8}, {10, 3667}, {40, 2957}, {517, 1647}, {5697, 6018}

X(14027) =  MIDPOINT OF X(2718) AND X(6788)

Barycentrics    (2 a-b-c)^2 (b-c)^2 (a+b-c) (a-b+c) : :

See Le Viet An, Peter Moses, and César Lozada, Hyacinthos 26382 and Hyacinthos 26383.

X(14027) lies on the incircle and these lines:
{8, 6079}, {11, 3667}, {55, 2743}, {56, 2222}, {57, 3322}, {214, 519}, {244, 1365}, {513, 1357}, {517, 6018}, {1155, 3021}, {1358, 3676}, {1362, 3660}, {1366, 1447}, {1397, 2720}, {1647, 3259}, {3328, 3675}, {4014, 5577}, {5061, 5211}, {5433, 6789}, {6790, 7288}

X(14027) = midpoint of X(2718) and X(6788)
X(14027) = crosspoint of X(3676) and X(3911)
. X(14027) = crosssum of X(i) and X(j) for these (i,j): {55, 5548}, {2316, 3939}
X(14027) = reflection of X(11) in the line X(1)X(2)
X(14027) = reflection of X(1357) in the line X(1)X(3)
X(14027) = X(i)-isoonjugate of X(j) for these (i,j): {644, 4638}, {679, 6065}, {765, 1318}, {1320, 9268}, {2316, 5376}, {3257, 5548}, {3939, 4618}
X(14027) = {X(244),X(6075)}-harmonic conjugate of X(7336)
X(14027) = barycentric product X(i)*X(j) for these {i,j}: {279, 4542}, {1086, 1317}, {1358, 4370}, {1647, 3911}, {3676, 6544}
X(14027) = barycentric quotient X(i)/X(j) for these {i,j}: {900, 4582}, {1015, 1318}, {1017, 6065}, {1317, 1016}, {1319, 5376}, {1357, 2226}, {1404, 9268}, {1647, 4997}, {1960, 5548}, {2087, 1320}, {3251, 644}, {3669, 4618}, {4370, 4076}, {4542, 346}, {4543, 6558}, {6544, 3699}

X(14028) =  X(1)X(2)∩X(11)X(11717)

Barycentrics    (2 a-b-c) (a^3-a^2 b-3 a b^2-b^3-a^2 c+7 a b c+b^2 c-3 a c^2+b c^2-c^3) : :

See Le Viet An, Peter Moses, and César Lozada, Hyacinthos 26382 and Hyacinthos 26383.

X(14028) lies on these lines: {1, 2}, {11, 11717}, {900, 1387}

X(14028) = midpoint of X(1) and X(1647)
X(14028) = {X(1),X(6788)}-harmonic conjugate of X(3244)

X(14029) =  (name pending)

Barycentrics    a^2 (a^11 b^3-5 a^9 b^5+10 a^7 b^7-10 a^5 b^9+5 a^3 b^11-a b^13-a^11 b^2 c-2 a^10 b^3 c+8 a^9 b^4 c+7 a^8 b^5 c-22 a^7 b^6 c-8 a^6 b^7 c+28 a^5 b^8 c+2 a^4 b^9 c-17 a^3 b^10 c+2 a^2 b^11 c+4 a b^12 c-b^13 c-a^11 b c^2+4 a^10 b^2 c^2-3 a^9 b^3 c^2-18 a^8 b^4 c^2+21 a^7 b^5 c^2+28 a^6 b^6 c^2-35 a^5 b^7 c^2-16 a^4 b^8 c^2+24 a^3 b^9 c^2-6 a b^11 c^2+2 b^12 c^2+a^11 c^3-2 a^10 b c^3-3 a^9 b^2 c^3+22 a^8 b^3 c^3-7 a^7 b^4 c^3-46 a^6 b^5 c^3+20 a^5 b^6 c^3+35 a^4 b^7 c^3-12 a^3 b^8 c^3-10 a^2 b^9 c^3+a b^10 c^3+b^11 c^3+8 a^9 b c^4-18 a^8 b^2 c^4-7 a^7 b^3 c^4+40 a^6 b^4 c^4+3 a^5 b^5 c^4-28 a^4 b^6 c^4-15 a^3 b^7 c^4+12 a^2 b^8 c^4+11 a b^9 c^4-6 b^10 c^4-5 a^9 c^5+7 a^8 b c^5+21 a^7 b^2 c^5-46 a^6 b^3 c^5+3 a^5 b^4 c^5+10 a^4 b^5 c^5+15 a^3 b^6 c^5+8 a^2 b^7 c^5-18 a b^8 c^5+5 b^9 c^5-22 a^7 b c^6+28 a^6 b^2 c^6+20 a^5 b^3 c^6-28 a^4 b^4 c^6+15 a^3 b^5 c^6-24 a^2 b^6 c^6+9 a b^7 c^6+4 b^8 c^6+10 a^7 c^7-8 a^6 b c^7-35 a^5 b^2 c^7+35 a^4 b^3 c^7-15 a^3 b^4 c^7+8 a^2 b^5 c^7+9 a b^6 c^7-10 b^7 c^7+28 a^5 b c^8-16 a^4 b^2 c^8-12 a^3 b^3 c^8+12 a^2 b^4 c^8-18 a b^5 c^8+4 b^6 c^8-10 a^5 c^9+2 a^4 b c^9+24 a^3 b^2 c^9-10 a^2 b^3 c^9+11 a b^4 c^9+5 b^5 c^9-17 a^3 b c^10+a b^3 c^10-6 b^4 c^10+5 a^3 c^11+2 a^2 b c^11-6 a b^2 c^11+b^3 c^11+4 a b c^12+2 b^2 c^12-a c^13-b c^13) : :

See Le Viet An and Peter Moses, Hyacinthos 26386.

X(14029) lies on ths line: {1, 3025}

leftri

Moses-Euler Points: X(14030)-X(14047)

rightri

On July 21, 2017, Peter Moses noted a relatively simple form for a point P(k) on the Euler line: P(k) = b4 - b2c2 + c4 + k*(a4 + b2c2) : : (barycentrics)

The point is here named the Moses-Euler Point (k = f), where f is a function of (a,b,c), homogeneous of degree 0. The point is given by the combo P(k) = 3(k + 1)(a4 + b4 + c4)*X(2) + 4(k - 1)S2*X(4).

Following is a list of examples:

P(-8) = X(14030)
P(-7) = X(14031)
P(-6) = X(14032)
P(-5) = X(14033)
P(-4) = X(14034)
P(-3) = X(14035)
P(-2) = X(11361)
P(-1) = X(4)
P(0) = X(5025)
P(1) = X(2)
P(2) = X(7892)
P(3) = X(14001)
P(4) = X(14036)
P(5) = X(14037)
P(6) = X(14038)
P(7) = X(14039)
P(8) = X(14040)
P(infinity) = X(384)
P(-1/2) = X(14041)
P(1/2) = X(7901)
P(-3/2) = X(14042)
P(3/2) = X(14043)
P(-3/4) = X(14044)
P(-1/4) = X(14045)
P(1/4) = X(14046)
P(3/4) = X(14047)
P(-2/3) = X(14062)
P(-1/3) = X(14063)
P(1/3) = X(14064)
P(2/3) = X(14065)
P(-4/3) =X(14066)
P(4/3) =X(14067)
P(-5/3) = X(14068)
P(5/3) = X(14069)

Every point on the Euler line is a Moses-Euler (k = f(a,b,c)) point. A few more examples follow:

If k = (4S^2-(a^4+b^4+c^4))/(4S^2+(a^4+b^4+c^4)), then P(k) = X(3).

If k = (4S^2+(a^4+b^4+c^4))/(4S^2-(a^4+b^4+c^4)), then P(k) = X(5).

If k = (2S^2-(a^4+b^4+c^4))/(2S^2+(a^4+b^4+c^4)), then P(k) = X(20).

If k = (12S^2-(a^4+b^4+c^4))/(12S^2+(a^4+b^4+c^4)), then P(k) = X(140).

If k = ((a+b+c) (a^3-a^2 b-a b^2+b^3-a^2 c-b^2 c-a c^2-b c^2+c^3)+(a^4+b^4+c^4))/((a+b+c) (a^3-a^2 b-a b^2+b^3-a^2 c-b^2 c-a c^2-b c^2+c^3)-(a^4+b^4+c^4)), then P(k) = X(21).

If k = (a^6-a^4 b^2-a^2 b^4+b^6-a^4 c^2-2 a^2 b^2 c^2-b^4 c^2-a^2 c^4-b^2 c^4+c^6+(a^2+b^2+c^2) (a^4+b^4+c^4))/(a^6-a^4 b^2-a^2 b^4+b^6-a^4 c^2-2 a^2 b^2 c^2-b^4 c^2-a^2 c^4-b^2 c^4+c^6-(a^2+b^2+c^2) (a^4+b^4+c^4)), then P(k) = X(22).

The Shinagawa coefficients (G(a,b,c), H(a,b,c)) for the Moses-Euler point P(k) = b4 - b2c2 + c4 + k*(a4 + b2c2) : : are given by

G(a,b,c) = (k + 1)( - 1 + cot2 ω)
H(a,b,c) = 2(k - 1).

Shinagawa coefficients are defined in the Introduction, in Part 1 of ETC. Related tables, also accessible at the top and bottom of each Part of ETC, are the following:

S-Coefficients (Euler Line, ++)
S-Coefficients (Euler Line, +-)
S-Coefficients, Midpoints on Euler Line, I
S-Coefficients, Midpoints on Euler Line, II
Midpoints Not on Euler Line

Following are combos for the Moses-Euler point P(k):

P(k) = 3 (a^4 + b^4 + c^4) (1 + k) X(2) + 4 (k - 1) S^2 X(4)
P(k) = 3 (1 + k) (Cot(w)^2 - 1) X(2) + 2 (k - 1) X(4)

Related combos for points on the Euler line:

X(2) + h X(3) = S^2 - 3 h /(2 + 3 h) SB SC : :
X(2) + h X(5) = S^2 + 3 h /(4 + 3 h) SB SC : :
X(3) + h X(4) = S^2 + (2 h - 1) SB SC : :
X(3) + h X(5) = S^2 + (h - 2) / (2 + h) SB SC : :

X(14030) =  MOSES-EULER POINT (k = -8)

Barycentrics    -8*a^4 + b^4 - 9*b^2*c^2 + c^4 : :

X(14030) lies on these lines: {2, 3}, {7823, 7896}, {7837, 12156}

X(14030) = {X(384),X(5025)}-harmonic conjugate of X(14040)

X(14031) =  MOSES-EULER POINT (k = -7)

Barycentrics    -7*a^4 + b^4 - 8*b^2*c^2 + c^4 : :

X(14031) lies on these lines: {2, 3}, {3734, 7826}, {7737, 7768}, {7747, 7869}, {7755, 11185}, {7795, 7860}, {7802, 10159}, {10352, 10992}

X(14031) = {X(384),X(5025)}-harmonic conjugate of X(14039)

X(14032) =  MOSES-EULER POINT (k = -6)

Barycentrics    -6*a^4 + b^4 - 7*b^2*c^2 + c^4 : :

X(14032) lies on these lines: {2, 3}, {598, 7863}, {3734, 7893}, {7747, 7922}, {7794, 7823}, {7854, 10302}

X(14032) = {X(384),X(5025)}-harmonic conjugate of X(14038)

X(14033) =  MOSES-EULER POINT (k = -5)

Barycentrics    -5*a^4 + b^4 - 6*b^2*c^2 + c^4 : :

X(14033) lies on these lines: {2, 3}, {69, 754}, {83, 7738}, {99, 7736}, {148, 8289}, {385, 1285}, {538, 1992}, {543, 5034}, {598, 7799}, {1007, 5475}, {1056, 4366}, {1058, 6645}, {1916, 8591}, {2386, 11188}, {2548, 6337}, {2549, 3618}, {2996, 5305}, {3053, 13468}, {3407, 5485}, {3619, 7761}, {3849, 5207}, {3926, 7745}, {3972, 7735}, {7622, 11147}, {7747, 7795}, {8716, 9300}, {10352, 13172}, {12176, 12243}

X(14033) = reflection of X(2) in X(11286)
X(14033) = reflection of X(32986) in X(2)
X(14033) = complement of X(33272)
X(14033) = anticomplement of X(11287)
X(14033) = orthocentroidal-circle-inverse of X(16041)
X(14033) = {X(2),X(4)}-harmonic conjugate of X(16041)
X(14033) = {X(2),X(20)}-harmonic conjugate of X(8356)
X(14033) = {X(384),X(5025)}-harmonic conjugate of X(14037)

X(14034) =  MOSES-EULER POINT (k = -4)

Barycentrics    -4*a^4 + b^4 - 5*b^2*c^2 + c^4 : :

X(14034) lies on these lines: {2, 3}, {148, 7920}, {316, 7869}, {543, 7878}, {598, 7764}, {1975, 7921}, {3314, 7747}, {3734, 7768}, {3972, 7755}, {5475, 7891}, {7737, 7893}, {7745, 7906}, {7748, 7875}, {7761, 10159}, {7777, 7816}, {7804, 7864}, {10351, 10796}

X(14034) = {X(2),X(4)}-harmonic conjugate of X(14045)
X(14034) = {X(384),X(5025)}-harmonic conjugate of X(14036)
X(14034) = orthocentroidal-circle-inverse of X(14045)

X(14035) =  MOSES-EULER POINT (k = -3)

Barycentrics    -3*a^4 + b^4 - 4*b^2*c^2 + c^4 : :

X(14035) lies on these lines: {2, 3}, {32, 11185}, {69, 7823}, {76, 7737}, {83, 2549}, {99, 2548}, {148, 4027}, {193, 732}, {315, 3734}, {316, 7795}, {388, 4366}, {497, 6645}, {543, 7772}, {598, 7858}, {671, 7856}, {1007, 7891}, {1916, 5395}, {1975, 7745}, {2996, 3407}, {3329, 7738}, {3618, 7864}, {3619, 7928}, {3620, 5104}, {3767, 3972}, {3849, 7854}, {3926, 7785}, {5319, 12150}, {5475, 7763}, {6337, 7777}, {6392, 7766}, {6781, 7815}, {7736, 7783}, {7739, 7878}, {7748, 7803}, {7753, 7781}, {7756, 7808}, {7758, 7812}, {7773, 7789}, {7775, 7863}, {7800, 7802}, {7801, 7843}, {7820, 7825}, {7822, 7842}, {7829, 11648}, {7872, 7889}, {10131, 10359}, {10352, 10358}

X(14035) = reflection of X(33263) in X(2)
X(14035) = complement of X(32997)
X(14035) = anticomplement X(7791)
X(14035) = {X(2),X(4)}-harmonic conjugate of X(14063)
X(14035) = {X(384),X(5025)}-harmonic conjugate of X(14001)
X(14035) = orthocentroidal-circle-inverse of X(14063)
X(14035) = {X(2),X(3)}-harmonic conjugate of X(33012)
X(14035) = {X(2),X(5)}-harmonic conjugate of X(33270)
X(14035) = {X(2),X(20)}-harmonic conjugate of X(32965)

X(14036) =  MOSES-EULER POINT (k = 4)

Barycentrics    4*a^4 + b^4 + 3*b^2*c^2 + c^4 : :

X(14036) lies on these lines: {2, 3}, {83, 7891}, {99, 7875}, {262, 11156}, {543, 7884}, {754, 3314}, {1916, 2482}, {1975, 7920}, {3407, 10302}, {3734, 7806}, {6781, 7937}, {7737, 7931}, {7745, 7945}, {7747, 7930}, {7753, 7870}, {7756, 7943}, {7759, 12156}, {7777, 7804}, {7782, 7889}, {7787, 7789}, {7795, 7893}, {7801, 7837}, {7802, 7915}, {7812, 7880}, {7816, 7846}, {7818, 7823}, {7822, 7904}, {7836, 7921}, {7863, 7878}, {8290, 9890}, {8724, 12176}

X(14036) = {X(2),X(4)}-harmonic conjugate of X(14046)
X(14036) = {X(384),X(5025)}-harmonic conjugate of X(14034)
X(14036) = orthocentroidal-circle-inverse of X(14046)

X(14037) =  MOSES-EULER POINT (k = 5)

Barycentrics    5*a^4 + b^4 + 4*b^2*c^2 + c^4 : :

X(14037) lies on these lines: {2, 3}, {193, 12212}, {315, 7820}, {1285, 7893}, {2548, 7835}, {2549, 7846}, {3329, 6337}, {3618, 7783}, {3619, 7904}, {3734, 7755}, {3926, 7787}, {3972, 7768}, {5286, 10583}, {6680, 11185}, {6781, 7914}, {7736, 7891}, {7737, 7832}, {7738, 7875}, {7758, 12150}, {7763, 7804}, {7774, 7789}, {7800, 10159}, {7803, 7816}, {10333, 10788}, {10353, 12176}


X(14038) =  MOSES-EULER POINT (k = 6)

Barycentrics    6*a^4 + b^4 + 5*b^2*c^2 + c^4 : :

X(14038) lies on these lines: {2, 3}, {3972, 7794}, {7789, 7921}, {7804, 7891}, {7816, 7875}, {7820, 7823}

X(14038) = {X(384),X(5025)}-harmonic conjugate of X(14032)

X(14039) =  MOSES-EULER POINT (k = 7)

Barycentrics    7*a^4 + b^4 + 6*b^2*c^2 + c^4 : :

X(14039) lies on these lines: {2, 3}, {69, 1285}, {83, 6337}, {99, 3618}, {597, 8716}, {754, 7795}, {1007, 7835}, {1992, 5039}, {3734, 7735}, {5149, 9890}, {5182, 12176}, {7736, 7804}, {7737, 7818}, {7738, 7816}, {7753, 9770}, {7757, 9741}, {7789, 9766}, {7796, 12156}

X(14039) = reflection of X(2) in X(33237)
X(14039) = reflection of X(33190) in X(2)
X(14039) = complement of X(33210)
X(14039) = {X(2),X(4)}-harmonic conjugate of X(33285)
X(14039) = {X(2),X(20)}-harmonic conjugate of X(11287)
X(14039) = orthocentroidal-circle-inverse of X(33285)
X(14039) = {X(384),X(5025)}-harmonic conjugate of X(14031)

X(14040) =  MOSES-EULER POINT (k = 8)

Barycentrics    8*a^4 + b^4 + 7*b^2*c^2 + c^4 : :

X(14040) lies on these lines: {2, 3}, {3972, 7826}, {7820, 7860}, {7823, 7869}, {7904, 10159}

X(14040) = {X(384),X(5025)}-harmonic conjugate of X(14030)

X(14041) =  MOSES-EULER POINT (k = -1/2)

Barycentrics    a4 - 2b4 - 2c4 + 3b2c2 : :
X(14041) = 2 X(115) + X(316) = X(148) + 2 X(325) = 4 X(115) - X(385) = 2 X(316) + X(385) = X(99) - 4 X(625) = 2 X(1513) - 5 X(3091) = 2 X(4) + X(5999) = 4 X(6722) - X(6781) = 2 X(671) + X(7840) = 2 X(99) - 5 X(7925) = 8 X(625) - 5 X(7925) = X(2) + 2 X(8352) = 5 X(2) - 8 X(8355) = 5 X(8352) + 4 X(8355) = 4 X(8352) - X(8597) = 2 X(2) + X(8597) = 16 X(8355) + 5 X(8597) = 5 X(2) - 2 X(8598) = 4 X(8355) - X(8598) = 5 X(8352) + X(8598) = 5 X(8597) + 4 X(8598) = 8 X(8598) - 5 X(9855) = 4 X(2) - X(9855) = 8 X(8352) + X(9855) = 2 X(8597) + X(9855) = 11 X(5056) - 8 X(10011) = 5 X(7948) - 2 X(10997) = 4 X(5) - X(11676) = 4 X(5103) - X(12215) = 2 X(6787) + X(13207) = X(98) + 2 X(13449) = 16 X(8355) - 5 X(13586) = 4 X(8598) - 5 X(13586) = 4 X(8352) + X(13586)

X(14041) lies on these lines: {2,3}, {76,7818}, {83,7861}, {98,13449}, {99,625}, {115,316}, {148,325}, {183,7898}, {194,7773}, {524,5207}, {538,671}, {543,7799}, {598,3407}, {1078,7842}, {1506,7847}, {1975,7912}, {2387,6787}, {2548,7864}, {2549,7777}, {3314,11185}, {3329,5475}, {3583,4366}, {3585,6645}, {3734,7931}, {3767,7823}, {3849,8859}, {3934,7911}, {3972,7844}, {5007,12156}, {5103,12215}, {5182,11645}, {5254,7785}, {5286,7921}, {5309,7812}, {6054,11152}, {6722,6781}, {7737,7806}, {7745,7797}, {7746,7802}, {7747,7828}, {7748,7752}, {7750,13468}, {7751,7860}, {7753,7827}, {7754,7900}, {7756,7769}, {7757,7775}, {7760,7843}, {7765,7858}, {7781,7814}, {7782,7862}, {7786,7872}, {7787,7851}, {7798,7926}, {7804,7919}, {7808,7918}, {7815,7910}, {7816,7899}, {7817,12150}, {7878,7902}, {7883,9466}, {8667,9939}, {9830,12151}

X(14041) = midpoint of X(i) and X(j) for these {i,j}: {671, 7809}, {8597, 13586}
X(14041) = reflection of X(i) in X(j) for these {i,j}: {2,33228}, {7840, 7809}, {8859, 9166}, {9855, 13586}, {13586, 2}
X(14041) = complement of X(33265)
X(14041) = orthocentroidal-circle-inverse of X(11361)
X(14041) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 4, 11361), (2, 3552, 11288), (2, 6655, 8356), (2, 7841, 7924), (2, 8352, 8597), (2, 8356, 7824), (2, 8597, 9855), (2, 11361, 384), (4, 5025, 384), (5, 6655, 7824), (5, 8356, 2), (76, 7825, 7885), (76, 7885, 7939), (83, 7861, 7923), (99, 625, 7925), (115, 316, 385), (194, 7773, 7941), (381, 7841, 2), (382, 7887, 3552), (384, 5025, 7901), (1003, 11318, 2), (1975, 7912, 7947), (3627, 7807, 6658), (3734, 7934, 7931), (3830, 11318, 1003), (3934, 7911, 7928), (5025, 11361, 2), (5254, 7785, 7839), (5475, 7790, 3329), (5999, 11361, 9855), (6661, 8360, 2), (7748, 7752, 7783), (7770, 7933, 7948), (7775, 11648, 7757), (7887, 11288, 2), (8355, 8598, 2)

X(14042) =  MOSES-EULER POINT (k = -3/2)

Barycentrics    3 a^4-2 b^4+5 b^2 c^2-2 c^4 : :
X(14042) = 3 (a^4+b^4+c^4) X(2) + 20 S^2 X(4)

X(14042) lies on these lines: {2,3}, {148,7745}, {316,7794}, {385,7747}, {543,7858}, {598,7772}, {671,5007}, {1975,7941}, {3329,7748}, {3583,6645}, {3585,4366}, {3631,5207}, {3734,7885}, {5475,7783}, {7773,7947}, {7804,7923}, {7816,7925}, {7823,11185}, {7825,7931}, {7840,7843}, {7842,7928}, {7878,11648}, {10131,10358}

X(14042) = {X(2),X(4)}-harmonic conjugate of X(14062)
X(14042) = {X(384),X(5025)}-harmonic conjugate of X(14043)
X(14042) = orthocentroidal-circle-inverse of X(14062)

X(14043) =  MOSES-EULER POINT (k = 3/2)

Barycentrics    3*a^4+2*b^4+b^2*c^2+2*c^4 : :
X(14043) = 15 (a^4+b^4+c^4) X(2) + 4 S^2 X(4)

X(14043) lies on these lines: {2,3}, {6,7945}, {32,7850}, {83,7874}, {99,7852}, {187,7928}, {325,10583}, {385,6680}, {574,7943}, {620,7859}, {1078,7915}, {1384,7929}, {1916,6683}, {1975,7932}, {3053,7938}, {3329,3788}, {3407,7815}, {3734,7942}, {3926,7920}, {3972,7867}, {5007,7840}, {5008,7917}, {5206,7937}, {6179,7869}, {7760,7880}, {7763,7875}, {7766,7881}, {7769,7889}, {7771,7914}, {7772,7870}, {7778,7787}, {7781,7884}, {7782,7913}, {7783,7834}, {7789,7797}, {7792,7836}, {7793,7868}, {7795,7806}, {7799,7829}, {7801,7856}, {7803,7891}, {7804,7899}, {7808,7940}, {7816,7919}, {7820,7828}, {7821,12150}, {7822,7857}, {7827,7863}, {7878,7888}, {7894,7908}, {8859,10302}

X(14043) = {X(2),X(4)}-harmonic conjugate of X(14065)
X(14043) = {X(384),X(5025)}-harmonic conjugate of X(14042)
X(14043) = orthocentroidal-circle-inverse of X(14065)

X(14044) =  MOSES-EULER POINT (k = -3/4)

Barycentrics    3 a^4-4 b^4+7 b^2 c^2-4 c^4 : :
X(14044) = 3 (a^4+b^4+c^4) X(2) - 28 S^2 X(4)

X(14044) lies on these lines: {2,3}, {148,7941}, {316,7826}, {598,7902}, {671,7843}, {3630,5207}, {7794,7885}, {7825,7922}, {7939,11185}

X(14044) = {X(2),X(4)}-harmonic conjugate of X(14066)
X(14044) = {X(384),X(5025)}-harmonic conjugate of X(14047)
X(14044) = orthocentroidal-circle-inverse of X(14066)

X(14045) =  MOSES-EULER POINT (k = -1/4)

Barycentrics    a^4-4 b^4+5 b^2 c^2-4 c^4 : :
X(14045) = 9 (a^4+b^4+c^4) X(2) - 20 S^2 X(4)

X(14045) lies on these lines: {2,3}, {115,7768}, {148,7947}, {316,7755}, {385,7825}, {625,7783}, {671,7821}, {3329,7861}, {3629,5207}, {5254,7941}, {5475,7923}, {7748,7925}, {7773,7839}, {7780,9166}, {7814,11648}, {7830,12815}, {7853,10159}, {7869,7934}

X(14045) = {X(2),X(4)}-harmonic conjugate of X(14034)
X(14045) = {X(384),X(5025)}-harmonic conjugate of X(14046)
X(14045) = orthocentroidal-circle-inverse of X(14034)

X(14046) =  MOSES-EULER POINT (k = 1/4)

Barycentrics    a^4+4 b^4-3 b^2 c^2+4 c^4 : :
X(14046) = 15 (a^4+b^4+c^4) X(2) - 12 S^2 X(4)

X(14046) lies on these lines: {2,3}, {115,7931}, {385,7818}, {597,5207}, {625,3329}, {671,7880}, {754,7828}, {1916,9166}, {2896,13468}, {3767,7939}, {5254,7947}, {5309,7840}, {6722,7831}, {7746,7928}, {7752,7923}, {7773,7932}, {7775,7884}, {7783,7861}, {7790,7925}, {7797,7941}, {7809,7817}, {7811,8859}, {7814,7902}, {7825,7942}, {7839,7851}, {7843,12156}, {7862,7918}, {7870,11648}, {7872,7940}, {7886,7911}

X(14046) = {X(2),X(4)}-harmonic conjugate of X(14036)
X(14046) = {X(384),X(5025)}-harmonic conjugate of X(14045)
X(14046) = orthocentroidal-circle-inverse of X(14036)

X(14047) =  MOSES-EULER POINT (k = 3/4)

Barycentrics    3*a^4+4*(b^2+c^2)^2-9*b^2*c^2 : :
X(14047) = 21 (a^4+b^4+c^4) X(2) - 4 S^2 X(4)

X(14047) lies on these lines: {2,3}, {385,7867}, {3329,7852}, {3788,7923}, {6680,7885}, {7778,7839}, {7783,7874}, {7792,7941}, {7794,7828}, {7797,7947}, {7806,7939}, {7817,7909}, {7834,7925}, {7840,7856}, {7844,7930}, {7851,7945}, {7854,8859}, {7857,7928}, {7862,7943}, {7870,7902}, {7884,7888}, {7886,7944}, {7913,7940}

X(14047) = {X(2),X(4)}-harmonic conjugate of X(14067)
X(14047) = {X(384),X(5025)}-harmonic conjugate of X(14044)
X(14047) = orthocentroidal-circle-inverse of X(14067)

X(14048) =  (name pending)

Barycentrics    a^7 + 5 a^6 (b+c) - a^5 (53 b^2-22 b c+53 c^2) - a^4 (29 b^3-129 b^2 c-129 b c^2+29 c^3) - a^3 (-79 b^4+64 b^3 c+222 b^2 c^2+64 b c^3-79 c^4) + a^2 (19 b^5-133 b^4 c+146 b^3 c^2+146 b^2 c^3-133 b c^4+19 c^5) -3 a (b^2-c^2)^2 (9 b^2-14 b c+9 c^2) + (b-c)^2 (b+c)^3 (5 b^2-6 b c+5 c^2) : :
X(14048) = 2r(13r-6R) X(40) - 3(4r(r+3R)-s^2) X(376)

See Tran Quang Hung and Angel Montesdeoca, Hyacinthos 26388.

X(14048) lies on this line: {40,376),

X(14049) =  (name pending)

Barycentrics    4 a^16-16 a^14 b^2+21 a^12 b^4-4 a^10 b^6-15 a^8 b^8+16 a^6 b^10-9 a^4 b^12+4 a^2 b^14-b^16-16 a^14 c^2+46 a^12 b^2 c^2-44 a^10 b^4 c^2+17 a^8 b^6 c^2-11 a^6 b^8 c^2+17 a^4 b^10 c^2-13 a^2 b^12 c^2+4 b^14 c^2+21 a^12 c^4-44 a^10 b^2 c^4+32 a^8 b^4 c^4-5 a^6 b^6 c^4-15 a^4 b^8 c^4+15 a^2 b^10 c^4-4 b^12 c^4-4 a^10 c^6+17 a^8 b^2 c^6-5 a^6 b^4 c^6+14 a^4 b^6 c^6-6 a^2 b^8 c^6-4 b^10 c^6-15 a^8 c^8-11 a^6 b^2 c^8-15 a^4 b^4 c^8-6 a^2 b^6 c^8+10 b^8 c^8+16 a^6 c^10+17 a^4 b^2 c^10+15 a^2 b^4 c^10-4 b^6 c^10-9 a^4 c^12-13 a^2 b^2 c^12-4 b^4 c^12+4 a^2 c^14+4 b^2 c^14-c^16 : :
X(14049) = 3 X(5642) - 4 X(11597) = 4 X(2914) - X(13202)

See Le Viet An and Peter Moses, Hyacinthos 26403.

X(14049) lies on these lines: {54,125}, {110,10112}, {113,137}, {539,5642}, {2777,12254}, {2888,5972}, {2914,12112}, {10212,10610}, {10619,10628}, {12121,12316}

X(14049) = midpoint of X(12121) and X(12316)
X(14049) = reflection of X(i) in X(j) for these {i,j}: {113, 11702}, {125, 54}, {2888, 5972}
X(14049) = {X(3043),X(10114)}-harmonic conjugate of X(125)

X(14050) =  (name pending)

Barycentrics    a^22-5 a^20 b^2+10 a^18 b^4-13 a^16 b^6+22 a^14 b^8-42 a^12 b^10+56 a^10 b^12-50 a^8 b^14+33 a^6 b^16-17 a^4 b^18+6 a^2 b^20-b^22-5 a^20 c^2+18 a^18 b^2 c^2-21 a^16 b^4 c^2+2 a^14 b^6 c^2+22 a^12 b^8 c^2-41 a^10 b^10 c^2+57 a^8 b^12 c^2-64 a^6 b^14 c^2+53 a^4 b^16 c^2-27 a^2 b^18 c^2+6 b^20 c^2+10 a^18 c^4-21 a^16 b^2 c^4+9 a^14 b^4 c^4+11 a^12 b^6 c^4-17 a^10 b^8 c^4-a^8 b^10 c^4+35 a^6 b^12 c^4-55 a^4 b^14 c^4+43 a^2 b^16 c^4-14 b^18 c^4-13 a^16 c^6+2 a^14 b^2 c^6+11 a^12 b^4 c^6+4 a^10 b^6 c^6-6 a^8 b^8 c^6-12 a^6 b^10 c^6+17 a^4 b^12 c^6-18 a^2 b^14 c^6+15 b^16 c^6+22 a^14 c^8+22 a^12 b^2 c^8-17 a^10 b^4 c^8-6 a^8 b^6 c^8+16 a^6 b^8 c^8+2 a^4 b^10 c^8-33 a^2 b^12 c^8-6 b^14 c^8-42 a^12 c^10-41 a^10 b^2 c^10-a^8 b^4 c^10-12 a^6 b^6 c^10+2 a^4 b^8 c^10+58 a^2 b^10 c^10+56 a^10 c^12+57 a^8 b^2 c^12+35 a^6 b^4 c^12+17 a^4 b^6 c^12-33 a^2 b^8 c^12-50 a^8 c^14-64 a^6 b^2 c^14-55 a^4 b^4 c^14-18 a^2 b^6 c^14-6 b^8 c^14+33 a^6 c^16+53 a^4 b^2 c^16+43 a^2 b^4 c^16+15 b^6 c^16-17 a^4 c^18-27 a^2 b^2 c^18-14 b^4 c^18+6 a^2 c^20+6 b^2 c^20-c^22 : :

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 26393.

X(14050) lies on this line: {128,10539}

X(14051) =  (name pending)

Barycentrics    24*S^4+2*(11*R^4-(5*SA+3*SW)* R^2-2*(5*SA+SW)*(SB+SC))*S^2-( 5*R^4+2*R^2*SW-4*SW^2)*(SB+SC) *SA : :
Trilinears    (cos 2A-cos 4A + 7/2)cos(B - C) + 2(cos A - cos 3A)cos(2B - 2 C) - 3(cos 2A - 1)cos(3B - 3C) + cos 5A + 3cos A - 3cos 3A : :

See Antreas Hatzipolakis and César Lozada, Hyacinthos 26410.

X(14051) lies on these lines: {5, 930}, {137, 5501}, {546, 1154}, {10285, 12026}

X(14051) = {X(137), X(5501)}-harmonic conjugate of X(8254)

X(14052) =  PERSPECTOR OF FEUERBACH HYPERBOLA OF THE ORTHIC TRIANGLE

Barycentrics    SB*SC /(5*S^2+SA^2+2*SB*SC-2*SW^2) : :
Barycentrics    (sec A)/(sin B sin C - e^2 sin(B + ω) sin(C + ω) - sin B sin(C + 2ω) - sin C sin(B + 2ω) + sin(B + 2ω) sin(C + 2ω)) : :

Let A'B'C' be the orthic triangle, and for arbitrary point P, let La be the line through A' parallel to AP, and define Lb and Lc cyclically. The locus of P such that La, Lb, Lc concur is the Jerabek hyperbola. The locus of the point of concurrence is the Feuerbach hyperbola of the orthic triangle, given by the barycentric equation

(SB - SC)(S2Ax2 + SBSCyz) + (SC - SA)(S2By2 + SCSAzx) + (SA - SB)(S2Cz2 + SASBxy) = 0.

The center of the hyperbola is X(1112), and the perspector is X(14052. The hyperbola passes through X(i) for these i: 4, 6, 52, 113, 155, 185, 193, 1162, 1163, 1829, 1839, 1843, 1858, 1986, 2574, 2575, 2904, 2905, 2906, 2907, 2914, 3574, 5095, 5895, 6152, 10294, 11817, 13202, 13420, 13431, as well and A' B' C' and the vertices of the cevian triangle of X(648).

See Antreas Hatzipolakis and César Lozada, Hyacinthos 26413.

Peter Moses (August 14, 2017) noted that this hyperbola appears in Example 2, page 145, in Clark Kimberling, "Conics associated with a cevian nest", Forum Geometricorum 1 (2001) 141-150.

The Hatzipolakis-Lozada hyperbola is the orthic isogonal conjugate of the Euler line, and the Feuerbach hyperbola of the orthic triangle if ABC is acute, and also the bicevian conic of X(4) and X(648). (Randy Hutson, November 2, 2017)

X(14052) lies on these lines: {468,34827}, {5186,16933}

X(14052) = isogonal conjugate of X(14060)
X(14052) = polar conjugate of X(14061)
X(14052) = barycentric product X(i)*X(j) for these {i, j}: {4, 36953}, {648, 36955}
X(14052) = barycentric quotient X(i)/X(j) for these (i, j): (4, 14061), (112, 33803), (115, 34953), (648, 33799), (811, 33809)
X(14052) = trilinear product X(i)*X(j) for these {i, j}: {19, 36953}, {162, 36955}
X(14052) = trilinear quotient X(i)/X(j) for these (i, j): (92, 14061), (162, 33803), (811, 33799), (1109, 34953)
X(14052) = trilinear pole of the line {14273, 36955}
X(14052) = intersection, other than A,B,C, of conics {{A, B, C, X(4), X(468)}} and {{A, B, C, X(25), X(460)}}
X(14052) = Cevapoint of X(25) and X(19504)
X(14052) = X(i)-isoconjugate-of-X(j) for these {i,j}: {48, 14061}, {63,39024}, {656, 33803}, {810, 33799}, {1101, 34953}
X(14052) = X(i)-reciprocal conjugate of-X(j) for these (i,j): (4, 14061), (112, 33803), (115, 34953), (648, 33799)

X(14053) =  X(4)X(916)∩X(6)X(31)

Barycentrics    a^2*(-a+b+c)*((b+c)*a^2+b*c*a- b^3-c^3)*((b^2+c^2)*a^3+b*c*( b+c)*a^2-(b^2-c^2)^2*a-(b^2-c^ 2)*(b-c)*b*c) : :

See Antreas Hatzipolakis and César Lozada, Hyacinthos 26413.

X(14053) lies on these lines: {4,916}, {6,31}, {185,516}, {1843,9028}, {2772,13202}

X(14054) =  X(1)X(6)∩X(4)X(912)

Barycentrics    a*((b+c)*a^2+2*b*c*a-(b^2-c^2) *(b-c))*(a^3-(b+c)*a^2-(b+c)^ 2*a+(b+c)*(b^2+c^2)) : :

See Antreas Hatzipolakis and César Lozada, Hyacinthos 26413.

X(14054) lies on these lines: {1,6}, {4,912}, {46,2900}, {65,3419}, {79,6598}, {145,6987}, {185,517}, {210,10198}, {226,3874}, {442,942}, {758,950}, {916,1902}, {1490,12704}, {1708,3811}, {2000,5707}, {2771,12690}, {2906,3193}, {2949,10902}, {3218,3651}, {3487,3873}, {3488,3869}, {3574,5777}, {3586,3901}, {3870,10267}, {3894,9612}, {3916,10391}, {3927,13615}, {4018,5895}, {4430,6846}, {4661,10587}, {5082,7672}, {5439,5705}, {5715,5927}, {5745,10122}, {5758,12116}, {6857,11020}, {6889,10202}, {7483,11018}, {11012,12675}

X(14054) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (9, 5904, 72), (72, 5728, 405), (1708, 3811, 11517), (5904, 10399, 9)

X(14055) =  X(4)X(5906)∩X(6)X(41)

Barycentrics    a^2*((b^2+c^2)*a^4+(b^3+c^3)* a^3-(b^3-c^3)*(b-c)*a^2-(b^3- c^3)*(b^2-c^2)*a-(b^2-c^2)^2* b*c)*((b+c)*a^4-b*c*a^3-(b+c)* (2*b^2-b*c+2*c^2)*a^2+b*c*(b+ c)^2*a+(b^3-c^3)*(b^2-c^2)) : :

See Antreas Hatzipolakis and César Lozada, Hyacinthos 26413.

X(14055) lies on these lines: {4,5906}, {6,41}, {185,515}

X(14056) =  COMPLEMENT OF X(3362)

Barycentrics    ((b+c)*a^5+b*c*a^4-2*(b^3+c^3) *a^3+(b^4-c^4)*(b-c)*a-(b^2-c^ 2)^2*b*c)*((b-c)^2*a^5+b*c*(b+ c)*a^4-2*(b^3-c^3)*(b-c)*a^3+( b^4-c^4)*(b^2-c^2)*a-(b^2-c^2) ^2*(b+c)*b*c) : :

See Antreas Hatzipolakis and César Lozada, Hyacinthos 26416.

X(14056) lies on these lines: {2,3362}, {1210,1785}

X(14056) = complement of X(3362)

X(14057) =  COMPLEMENT OF X(13855)

Barycentrics    (16*R^4+8*(SA-SW)*R^2-S^2-2*SA ^2+SW^2)*(2*S^4+(8*(-2*SA+SW)* R^2+(SA+SW)*(3*SA-2*SW))*S^2+( -SW^2+16*R^4)*(SA-SW)*SA) : :

See Antreas Hatzipolakis and César Lozada, Hyacinthos 26416.

X(14057) lies on this line: (2,13855)

X(14057) = complement of X(13855)

X(14058) =  COMPLEMENT OF X(1745)

Barycentrics    (b-c)^2*a^5-b*c*(b+c)*a^4-2*(b ^3-c^3)*(b-c)*a^3+(b^4-c^4)*(b ^2-c^2)*a+(b^2-c^2)^2*(b+c)*b* c : :

See Antreas Hatzipolakis and César Lozada, Hyacinthos 26416.

X(14058) lies on these lines: {2,1745}, {3,10}, {29,58}, {57,1940}, {1413,8808}

X(14058) = complement of X(1745)

X(14059) =  COMPLEMENT OF X(1075)

Barycentrics    SA*(SB+SC)*(16*R^4-8*(2*SA+SW) *R^2+S^2+(SA+SW)^2+SA^2) : :
X(14059) = X(3) + 2*X(8798)

See Antreas Hatzipolakis and César Lozada, Hyacinthos 26416.

X(14059) lies on these lines: {2,1075}, {3,64}, {4,2972}, {394,2055}, {417,11459}, {418,7999}, {852,11412}, {1216,6638}, {3090,13409}

X(14059) = complement of X(1075)
X(14059)= {X(6509), X(11793)}-harmonic conjugate of X(3)

X(14060) =  X(2)X(1632)∩X(3)X(895)

Barycentrics    a^2 (a^2-b^2-c^2) (a^4-a^2 b^2+2 b^4-a^2 c^2-3 b^2 c^2+2 c^4) : :

See Antreas Hatzipolakis and César Lozada, Hyacinthos 26413.

X(14060) lies on these lines: {2,1632}, {3,895}, {50,11416}, {74,9160}, {110,7669}, {216,248}, {566,5012}, {1634,9142}, {1992,9737}, {2407,11596}, {3563,6036}, {8553,12220}, {9019,11063}, {9512,12042}

X(14060) = isogonal conjugate of X(14052)
X(14060) = isotomic conjugate of polar conjugate of X(39024)
X(14060) = X(19)-isoconjugate of X(36953)
X(14060)= barycentric product X(3)*X(14061)

X(14061) =  X(2)X(99)∩X(5)X(83)

Trilinears    sin B sin C - e^2 sin(B + ω) sin(C + ω) - sin B sin(C + 2ω) - sin C sin(B + 2ω) + sin(B + 2ω) sin(C + 2ω) : :
Barycentrics    a^4-a^2 b^2+2 b^4-a^2 c^2-3 b^2 c^2+2 c^4 : :
X(14061) = 4 X(5) + X(98) = 6 X(2) - X(99) = 3 X(2) + 2 X(115) = X(99) + 4 X(115) = 6 X(115) - X(148) = 9 X(2) + X(148) = 3 X(99) + 2 X(148) = 3 X(99) - 8 X(620) = 9 X(2) - 4 X(620) = 3 X(115) + 2 X(620) = X(148) + 4 X(620) = 4 X(148) - 9 X(671) = 8 X(115) - 3 X(671) = 4 X(2) + X(671) = 2 X(99) + 3 X(671) = 16 X(620) + 9 X(671) = X(76) + 4 X(2023) = 7 X(99) - 12 X(2482) = 14 X(620) - 9 X(2482) = 7 X(2) - 2 X(2482)

X(14061) lies on these lines: {2,99}, {3,10723}, {4,6036}, {5,83}, {8,11725}, {10,7983}, {13,6670}, {14,6669}, {76,2023}, {114,3090}, et al

X(14061) = isotomic conjugate of X(36953)
X(14061) = anticomplement of X(31274)
X(14061) = crosssum of X(i) and X(j) for these (i,j): {32, 9696}, {2028, 2029}
X(14061) = polar conjugate of X(14052)
X(14061) = barycentric quotient X(14060)/X(3)
X(14061) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 115, 99), (2, 148, 620), (2, 5461, 9166), (2, 7844, 7919), (2, 8591, 9167), (2, 9166, 671), (2, 11185, 7835), (5, 7828, 83), (76, 7887, 7899), (76, 7899, 7909), (99, 115, 671), (99, 9166, 115), (115, 620, 148), (115, 6722, 2), (148, 620, 99), (5461, 6722, 115),.. .

X(14062) =  MOSES-EULER POINT (k = -2/3)

Barycentrics    2 (a^4+b^2 c^2)-3 (b^4-b^2 c^2+c^4) : :
X(14062) = 3 (a^4+b^4+c^4) X(2)-20 S^2 X(4)

X(14062) lies on these lines: {2,3}, {115,6179}, {148,7773}, {316,7751}, {543,7814}, {598,7829}, {625,7891}, {671,7759}, {3314,7825}, {5254,7921}, {5475,7864}, {7745,7920}, {7747,7806}, {7748,7777}, {7837,7843}, {7842,7904}, {7858,11648}, {7861,7875}, {7885,11185}, {9863,13449}, {10104,10242}

X(14062) = {X(2),X(4)}-harmonic conjugate of X(14042)
X(14062) = {X(384),X(5025)}-harmonic conjugate of X(14065)
X(14062) = orthocentroidal-circle-inverse of X(14042)

X(14063) =  MOSES-EULER POINT (k = -1/3)

Barycentrics    a^4+b^2 c^2-3 (b^4-b^2 c^2+c^4) : :
X(14063) = 3 (a^4+b^4+c^4) X(2)-8 S^2 X(4)

X(14063) lies on these lines: {2,3}, {69,7885}, {115,315}, {148,3926}, {193,5111}, {316,3767}, {543,7888}, {625,7748}, {626,11185}, {671,7796}, {1007,7783}, {1506,7872}, {1916,2996}, {2548,7790}, {2549,7752}, {3407,5395}, {3618,7923}, {3785,7898}, {4366,5225}, {5206,6722}, {5229,6645}, {5254,7773}, {5286,7785}, {5309,7843}, {5319,7812}, {5475,7803}, {6337,7925}, {6392,7779}, {6653,7080}, {7615,7883}, {7694,12203}, {7735,7823}, {7736,7864}, {7737,7828}, {7738,7777}, {7739,7858}, {7745,7851}, {7746,7842}, {7747,7844}, {7753,7902}, {7756,7862}, {7758,7809}, {7764,11648}, {7765,7775}, {7795,7934}, {7800,7911}, {9607,11163}

X(14063) = complement of X(33244)
X(14063) = anticomplement of X(16925)
X(14063) = {X(2),X(3)}-harmonic conjugate of X(33206)
X(14063) = {X(2),X(5)}-harmonic conjugate of X(33009)
X(14063) = {X(2),X(20)}-harmonic conjugate of X(32964)
X(14063) = {X(2),X(4)}-harmonic conjugate of X(14035)
X(14063) = {X(384),X(5025)}-harmonic conjugate of X(14064)
X(14063) = orthocentroidal-circle-inverse of X(14035)

X(14064) =  MOSES-EULER POINT (k = 1/3)

Barycentrics    2 (a^4+b^2 c^2)+3 (b^4-b^2 c^2+c^4) : :
X(14064) = 3 (a^4+b^4+c^4) X(2)-2 S^2 X(4)

X(14064) lies on these lines: {2,3}, {39,1007}, {69,626}, {115,7795}, {141,5490}, {148,7945}, {193,5305}, {230,3785}, {315,6179}, {316,7942}, {325,5286}, {620,7872}, {625,1692}, {1285,7823}, {1506,7913}, {1992,5319}, {2549,3788}, {3619,3934}, {3926,5254}, {3933,6392}, {4045,7862}, {4766,5230}, {5033,7808}, {5304,7762}, {5309,7758}, {5346,7845}, {5355,7903}, {5475,7852}, {6680,7737}, {6722,7815}, {7612,10104}, {7710,12203}, {7736,7752}, {7738,7763}, {7739,7764}, {7746,7800}, {7748,7874}, {7749,7935}, {7755,7818}, {7761,7886}, {7765,7888}, {7769,7918}, {7773,7792}, {7774,7797}, {7775,7829}, {7777,7923}, {7785,7932}, {7805,11008}, {7806,7885}, {7809,7856}, {7814,7827}, {7832,11185}, {7847,7940}, {7857,7911}, {7858,7884}, {7863,11648}, {7864,7925}, {7920,7941}, {9606,11184}

X(14064) = reflection of X(33224) in X(2)
X(14064) = complement of X(32973)
X(14064) = anticomplement of X(32954)
X(14064) = {X(2),X(3)}-harmonic conjugate of X(32970)
X(14064) = {X(2),X(5)}-harmonic conjugate of X(32968)
X(14064) = {X(2),X(20)}-harmonic conjugate of X(7807)
X(14064) = {X(2),X(4)}-harmonic conjugate of X(14001)
X(14064) = {X(384),X(5025)}-harmonic conjugate of X(14063)
X(14064) = orthocentroidal-circle-inverse of X(14001)

X(14065) =  MOSES-EULER POINT (k = 2/3)

Barycentrics    2 (a^4+b^2 c^2)+3 (b^4-b^2 c^2+c^4) : :
X(14065) = 15 (a^4+b^4+c^4) X(2)-4 S^2 X(4)

X(14065) lies on these lines: {2,3}, {115,7930}, {230,7938}, {325,7920}, {620,7918}, {625,7846}, {626,6179}, {1506,7943}, {1916,6722}, {3096,7886}, {3314,7751}, {3767,7931}, {3788,7864}, {4045,7940}, {5254,7945}, {5286,7947}, {5305,7897}, {5306,7946}, {5309,7909}, {5319,7840}, {5346,7917}, {5355,7871}, {5368,7949}, {6680,7823}, {7735,7939}, {7746,7944}, {7749,7937}, {7752,7852}, {7755,7922}, {7763,7923}, {7764,7884}, {7765,7870}, {7769,7913}, {7773,10583}, {7777,7834}, {7778,7797}, {7790,7874}, {7792,7912}, {7796,7817}, {7799,7902}, {7803,7925}, {7814,7829}, {7821,7837}, {7827,7888}, {7832,7844}, {7835,7861}, {7836,7851}, {7853,7857}, {7859,7862}

X(14065) = {X(2),X(4)}-harmonic conjugate of X(14043)
X(14065) = {X(384),X(5025)}-harmonic conjugate of X(14062)
X(14065) = orthocentroidal-circle-inverse of X(14043)

X(14066) =  MOSES-EULER POINT (k = -4/3)

Barycentrics    4 (a^4+b^2 c^2)-3 (b^4-b^2 c^2+c^4) : :
X(14066) = 3(a^4+b^4+c^4) X(2)+28 S^2 X(4)

X(14066) lies on these lines: {2,3}, {148,7921}, {316,7896}, {598,7765}, {6179,7747}, {7751,7823}, {7893,11185}

X(14066) = {X(2),X(4)}-harmonic conjugate of X(14044)
X(14066) = {X(384),X(5025)}-harmonic conjugate of X(14067)
X(14066) = orthocentroidal-circle-inverse of X(14044)

X(14067) =  MOSES-EULER POINT (k = 4/3)

Barycentrics    4 (a^4+b^2 c^2)+3 (b^4-b^2 c^2+c^4) : :
X(14067) = 21 (a^4+b^4+c^4) X(2)+4 S^2 X(4)

X(14067) lies on these lines: {2,3}, {620,7943}, {3314,6179}, {3788,7875}, {7751,7806}, {7777,7846}, {7778,7921}, {7789,7932}, {7792,7906}, {7820,7942}, {7823,7867}, {7829,7870}, {7834,7891}, {7835,7852}, {7836,7920}, {7837,7909}, {7856,7880}, {7857,7915}, {7863,7884}, {7889,7940}, {7893,7931}, {7904,7944}

X(14067) = {X(2),X(4)}-harmonic conjugate of X(14047)
X(14067) = {X(384),X(5025)}-harmonic conjugate of X(14066)
X(14067) = orthocentroidal-circle-inverse of X(14047)

X(14068) =  MOSES-EULER POINT (k = -5/3)

Barycentrics    5 (a^4+b^2 c^2)-3 (b^4-b^2 c^2+c^4) : :
X(14068) = 3 (a^4+b^4+c^4) X(2)+16 S^2 X(4)

X(14068) lies on these lines: {2,3}, {671,5319}, {2996,7766}, {4366,5229}, {5225,6645}, {6179,7737}, {7747,7751}

X(14068) = complement of X(33209)
X(14068) = anticomplement of X(32965)
X(14068) = {X(2),X(4)}-harmonic conjugate of X(32996)
X(14068) = orthocentroidal-circle-inverse of X(32996)

X(14069) =  MOSES-EULER POINT (k = 5/3)

Barycentrics    5 (a^4+b^2 c^2)+3 (b^4-b^2 c^2+c^4) : :
X(14069) = 3 (a^4+b^4+c^4) X(2)+ S^2 X(4)

X(14069) lies on these lines: {2,3}, {69,6179}, {83,1007}, {193,7881}, {315,1285}, {1078,3619}, {1992,7796}, {2548,7874}, {2549,7852}, {3618,7763}, {3767,7820}, {3785,7868}, {3788,7736}, {3926,7792}, {3933,5304}, {5286,7789}, {5319,7801}, {6337,7803}, {6680,7735}, {7737,7867}, {7738,7834}, {7739,7863}, {7758,7880}, {7774,7945}, {7800,7915}, {7827,9741}, {7888,9770}, {7942,11185}

X(14069) = complement of X(33180)
X(14069) = {X(2),X(3)}-harmonic conjugate of X(32956)
X(14069) = {X(2),X(4)}-harmonic conjugate of X(32951)
X(14069) = {X(2),X(5)}-harmonic conjugate of X(32955)
X(14069) = {X(2),X(20)}-harmonic conjugate of X(7866)
X(14069) = orthocentroidal-circle-inverse of X(32951)

X(14070) =  LEVERSHA POINT (EULER LINE INTERCEPT OF X(159)X(542))

Barycentrics    (SB+SC)*(2*SA^2-6*R^2*SA+2*S^2-3*SB*SC) : :
X(14070) = X(3)+2*X(26) = X(3)-4*X(1658) = 2*X(3)+X(7387) = 5*X(3)+7*X(10244) = X(3)+3*X(10245) = 7*X(3)-4*X(11250) = 5*X(3)-2*X(12084) = 4*X(3)-X(12085) = 4*X(156)-X(12164) = 2*X(2931)+X(12412) = X(2931)+2*X(13289) = X(12412)-4*X(13289)

As a point on the Euler lime, X(14070) has Shinagawa coefficients (-E-6*F, 3*E+6*F)

The insimilcenter of the circumcircles of the Kosnita and tangential triangles is referred as the Leversha point in the International Journal of Computer Discovered Mathematics. (César Lozada, July 25, 2017)

X(14070) lies on these lines: {2,3}, {35,9645}, {49,12160}, {143,11426}, {154,13754}, {155,10282}, {156,12164}, {159,542}, {511,11202}, {539,2917}, {541,10117}, {567,9777}, {568,11402}, {1154,3167}, {1498,7689}, {1511,10752}, {1989,2079}, {1993,11464}, {2782,9876}, {3311,11266}, {3312,11265}, {3654,8193}, {3679,9590}, {3796,9730}, {3964,6148}, {5050,5946}, {5085,5892}, {5446,11425}, {5654,10192}, {5655,12168}, {5889,9707}, {5890,6800}, {5944,12161}, {6759,12163}, {7753,9699}, {8276,13846}, {8277,13847}, {8989,12973}, {9300,9608}, {9659,10037}, {9672,10046}, {9826,12220}, {9833,12359}, {9908,9932}, {9921,13680}, {9922,13800}, {10056,10831}, {10072,10832}, {11216,11649}, {11267,11486}, {11268,11485}, {11820,12041}

X(14070) = midpoint of X(3) and X(9909)
X(14070) = reflection of X(i) in X(j) for these (i,j): (381,10201), (5654,10192), (7387,9909), (9909,26)
X(14070) = circumcircle-inverse-of-X(10297)
X(14070) = insimilcenter of the circumcircles of the Kosnita and tangential triangles (the exsimilcenter is X(3))
X(14070) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 382, 3516), (3, 2937, 11414), (3, 7387, 12085), (3, 7517, 1593), (22, 186, 3), (23, 13620, 10298), (3515, 9715, 3), (6644, 7502, 3), (7502, 7575, 6644), (9909, 10245, 26), (9915,9916,159), (10244, 12084, 7387)

X(14071) =  MIDPOINT OF X(195) AND X(930)

Trilinears    24*S^4+2*(11*R^4-(5*SA+3*SW)* R^2-2*(5*SA+SW)*(SB+SC))*S^2-( 5*R^4+2*R^2*SW-4*SW^2)*(SB+SC) *SA : :
Barycentrics    (2*a^10-7*(b^2+c^2)*a^8+10*(b^4+b^2*c^2+c^4)*a^6-(b^2+c^2)*(8*b^4-7*b^2*c^2+8*c^4)*a^4+(b^2-c^2)^2*(4*b^4+3*b^2*c^2+4*c^4)*a^2-(b^4-c^4)*(b^2-c^2)^3)*(a^6-2*(b^2+c^2)*a^4+(b^4+c^4-(b^2+b*c+c^2)*b*c)*a^2+(b^2-c^2)^2*b*c)*(a^6-2*(b^2+c^2)*a^4+(b^4+c^4+(b^2-b*c+c^2)*b*c)*a^2-(b^2-c^2)^2*b*c) : :

See Antreas Hatzipolakis and César Lozada, Hyacinthos 26420.

X(14071) lies on these lines: {5,6343}, {125,128}, {137,5501}, {195,930}, {6150,6592}

X(14071) = midpoint of X(i) and X(j) for these {i,j}: {5,6343}, {195,930}z
X(14071) = reflection of X(137) in X(8254)

X(14072) =  REFLECTION OF X(5) IN X(128)

Barycentrics    (S^2+SB*SC)*(4*SA^2+(6*R^2-4* SW)*SA-22*R^2*SW+4*SW^2+27*R^ 4) : :
X(14072) = 3*X(5)-2*X(137) = 3*X(128)-X(137) = 4*X(128)-X(1263) = 4*X(137)-3*X(1263) = 3*X(381)-X(11671)

See Le Viet An and César Lozada, Hyacinthos 26425.

X(14072) lies on the cubics K465 and K725 and on these lines: {2,12026}, {3,2888}, {4,13512}, {5,128}, {30,930}, {140,1141}, {381,11671}, {495,3327}, {496,7159}, {539,6150}, {549,13372}, {6069,14050}

X(14072) = midpoint of X(i) and X(j) for these {i,j}: {4,13512}, {6069,14050}
X(14072) = reflection of X(i) in X(j) for these (i,j): (3,6592), (5,128), (1141,140), (1263,5)
X(14072) = anticomplement of X(12026)

X(14073) =  REFLECTION OF X(1263) IN X(128)

Barycentrics    (S^2+SB*SC)*(8*SA^2+(12*R^2-8* SW)*SA+45*R^4+2*S^2+6*SW^2-36* R^2*SW) : :
X(14073) = 3*X(5)-4*X(128) = 5*X(5)-4*X(137) = 3*X(5)-2*X(1263) = 5*X(128)-3*X(137) = 6*X(137)-5*X(1263) = 3*X(549)-2*X(1141) = 3*X(549)-4*X(6592) = 5*X(632)-4*X(12026)

See Le Viet An and César Lozada, Hyacinthos 26425.

X(14073) lies on these lines: {5,128}, {30,13512}, {74,550}, {546,11671}, {549,1141}, {632,12026}, {2888,10205}

X(14073) = reflection of X(i) in X(j) for these (i,j): (550,930), (1141,6592), (1263,128), (11671,546)
X(14073) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (128, 1263, 5), (1141, 6592, 549)

X(14074) =  X(104)X(1001)∩X(105)X(999)

Barycentrics    a*(a-b)*(a-c)/(a^2-2*(b+c)* a+b^2+4*b*c+c^2) : :

See César Lozada, Hyacinthos 26432.

X(14074) occurs in connection with X(1156);/ see Ngo Quang Duong, ADGEOM #2795 (9/16/2015) and Angel Montesdeoca, ADGEOM #2800.

Let A'B'C' be the excentral triangle. The lines X(2)X(6) of triangles A'BC, B'CA, C'AB bound a triangle perspective to ABC at X(14074). (Randy Hutson, June 27, 2018)

X(14074) lies on the circumcircle and these lines: {1,2291}, {56,3321}, {103,3576}, {104,1001}, {105,999}, {106,7290}, {692,2742}, {840,5126}, {972,3428}, {1308,1633}, {1477,13462}, {2371,5223}, {2717,5144}

X(14074) = isogonal conjugate of X14077)
X(14074) = trilinear pole of X(6)X(1155)
X(14074) = X(126)-of-hexyl-triangle
X(14074) = X(5512)-of-excentral-triangle
X(14074) = Λ(normal to Feuerbach hyperbola at X(7))
X(14074) = Ψ(X(i), X(j)) for these (i,j): (1, 527), (6, 1155)
X(14074) = trilinear product of circumcircle intercepts of line X(1)X(527)

X(14075) =  X(3)X(6)∩X(111)X(7954)

Barycentrics    (7*a^2+4*(b^2+c^2))*a^2 : :
X(14075) = 3*S^2*X(3)-11*SW^2*X(6)

See Tran Quang Hung and César Lozada, Hyacinthos 26435.

X(14075) lies on these lines: {3,6}, {111,7954}, {1992,7820}, {3619,7826}, {3620,7889}, {3630,7822}, {3856,5305}, {5304,7753}, {5306,10109}, {5354,8585}, {7794,11008}, {7798,12150}, {7916,10583}

X(14075) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6, 187, 7772), (6, 5008, 574), (6, 5024, 5041), (39, 5210, 574), (574, 5008, 32)

X(14076) =  X(3)X(161)∩X(54)X(125)

Trilinears    cos(2A) cos(B - C) + (cos 3A) - cos(2B - 2C) - (1 + cos 2A) cos(3B - 3C) : :

See Antreas Hatzipolakis and César Lozada, Hyacinthos 26438.

X(14076) lies on these lines: {2,10274}, {3,161}, {5,10628}, {54,125}, {427,11808}, {575,8254}, {1092,2888}, {1154,5449}, {3153,7691}, {3567,3574}, {5498,12038}, {5965,8548}

X(14076) = complement of X(10274)

X(14077) =  ISOGONAL CONJUGATE OF X(14074)

Barycentrics    a (b-c) (a^2-2 a b+b^2-2 a c+4 b c+c^2) : :

X(14077) lies on these lines: {1,650}, {8,693}, {10,4885}, {30,511}, {72,4024}, {885,1000}, {905,4041}, {1002,2401}, {1022,11525}, {1387,10006}, {1734,3669}, {2254,4814}, {3057,11247}, {3126,4825}, {3295,8641}, {3696,4411}, {3803,4498}, {4040,4162}, {4297,8142}, {4378,4730}, {4474,4804}, {4490,4879}, {4724,4895}, {10246,11124}

X(14077) = isogonal conjugate of X14074)
X(14077) = crossdifference of every pair of points on line X(6)X(1155)
X(14077) = barycentric product X(i)*X(j) for these {i,j}: {522, 8545}, {1996, 3900} X(14077) = barycentric quotient X(i)/X(j) for these {i,j}: {1996, 4569}, {8545, 664}

X(14078) =  BSS(a → |b - c|) of X(1)

Barycentrics    |b - c| : |c - a|: |a - b|
Barycentrics    area(AIG) : area(BIG) : area(CIG), where I = X(1), G = X(2)

The mapping BSS(a → |b - c|), is an example of barycentric symbolic substitution, which is analogous to trilinear symbolic substitution; see X(3211).

X(14078) is the diagonal crosspoint of the cyclic quadrilateral whose vertices are the intersections of the incircle and Steiner inellipse; see X(5997). (Randy Hutson, November 2, 2017) X(14078) lies on this line: {9,14086}

X(14078) = isogonal conjugate of X(14085)
X(14078) = isotomic conjugate of X(14087)
X(14078) = barycentric square root of X(1086)
X(14078) = X(14087)-daleth conjugate of X(14078)
X(14078) = isoconjugate of X(j) and X(j) for these (i,j): {1, 14085}, {31, 14087}, {213, 14089}, {765, 14088}, {1110, 14078}, {1252, 14079}, {4567, 14090}
X(14078) = barycentric product X(i)*X(j) for these {i,j}: {1, 14080}, {75, 14079}, {76, 14088}, {86, 14086}, {145, 4373}, {145, 4859}, {310, 14090}, {1086, 14087}, {3120, 14089}
X(14078) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 14087}, {6, 14085}, {86, 14089}, {244, 14079}, {1015, 14088}, {1111, 14080}, {3120, 14086}, {3122, 14090}, {14079, 1}, {14080, 75}, {14085, 1252}, {14086, 10}, {14087, 1016}, {14088, 6}, {14089, 4600}, {14090, 42}

X(14079) =  BARYCENTRIC PRODUCT X(1)*X(14078)

Trilinears    area(AIG) : area(BIG) : area(CIG), where I = X(1), G = X(2)
Barycentrics    a|b - c| : b|c - a|: c|a - b| : :

In the plane of a triangle ABC, let A'B'C' and A''B''C'' be the intouch and extouch triangles. Then X(14079) has trilinears |A'A''| : |B'B''| : |C'C''|. (Randy Hutson, July 28, 2017)

X(14079) lies on these lines: {37,15222}, {1125,14083}, {3756,14084}, {14078,14086}

X(14079) = isogonal conjugate of the isotomic conjugate of X(14080)
X(14079) = barycentric product X(i)*X(j) for these {i, j}: {1, 14078}, {6, 14080}, {75, 14088}, {81, 14086}, {244, 14087}, {274, 14090}
X(14079) = barycentric quotient X(i)/X(j) for these (i, j): (1, 14087), (31, 14085), (81, 14089), (244, 14078), (1015, 14079), (1086, 14080)
X(14079) = trilinear product X(i)*X(j) for these {i, j}: {2, 14088}, {6, 14078}, {31, 14080}, {58, 14086}, {86, 14090}, {1015, 14087}
X(14079) = trilinear quotient X(i)/X(j) for these (i, j): (2, 14087), (6, 14085), (86, 14089), (244, 14079), (1015, 14088), (1086, 14078)
X(14079) = X(i)-isoconjugate-of-X(j) for these {i,j}: {2, 14085}, {6, 14087}, {42, 14089}
X(14079) = X(i)-reciprocal conjugate of-X(j) for these (i,j): (1, 14087), (31, 14085), (81, 14089), (244, 14078)
X(14079) = {X(15222), X(15223)}-harmonic conjugate of X(37)

X(14080) =  BARYCENTRIC PRODUCT X(75)*X(14078)

Barycentrics    bc|b - c| : ca|c - a|: ab|a - b| : :

X(14080) lies on these lines: {}

X(14080) = isotomic conjugate of the isogonal conjugate of X(14079)
X(14080) = barycentric product X(i)*X(j) for these {i, j}: {75, 14078}, {76, 14079}, {274, 14086}, {561, 14088}, {1111, 14087}
X(14080) = barycentric quotient X(i)/X(j) for these (i, j): (1, 14085), (75, 14087), (244, 14088), (274, 14089), (1086, 14079), (1111, 14078)
X(14080) = trilinear product X(i)*X(j) for these {i, j}: {2, 14078}, {75, 14079}, {76, 14088}, {86, 14086}, {310, 14090}, {1086, 14087}
X(14080) = trilinear quotient X(i)/X(j) for these (i, j): (2, 14085), (76, 14087), (310, 14089), (1086, 14088), (1111, 14079)
X(14080) = X(i)-isoconjugate-of-X(j) for these {i,j}: {6, 14085}, {32, 14087}, {1110, 14079}, {1252, 14088}
X(14080) = X(i)-reciprocal conjugate of-X(j) for these (i,j): (1, 14085), (75, 14087), (244, 14088), (274, 14089)

X(14081) =  SINGULAR FOCUS OF THE CUBIC K066a

Barycentrics    2 SA (S (2 S+Sqrt(3) SA)-SB SC)^2-(S (2 S+Sqrt(3) SA)-SB SC) ((-b^2+2 c^2) (S (2 S+Sqrt(3) SB)-SA SC)+(2 b^2-c^2) (-SA SB+S (2 S+Sqrt(3) SC)))+a^2 ((S (2 S+Sqrt(3) SB)-SA SC)^2-(S (2 S+Sqrt(3) SB)-SA SC) (-SA SB+S (2 S+Sqrt(3) SC))+(-SA SB+S (2 S+Sqrt(3) SC))^2) : :

X(14081) lies on the curve Q041 and these lines: {2, 5469}, {137, 630}, {635, 2380}

X(14081) = orthoptic-circle-of-Steiner-inellipe-inverse of X(5982)
X(14081) = circumcircle-of-outer-Napoleon-triangle-inverse of X(14144)
X(14081) = psi-transform of X(627)

X(14082) =  SINGULAR FOCUS OF THE CUBIC K066b

Barycentrics    2 SA (-S (-2 S+Sqrt(3) SA)-SB SC)^2-(-S (-2 S+Sqrt(3) SA)-SB SC) ((-b^2+2 c^2) (-S (-2 S+Sqrt(3) SB)-SA SC)+(2 b^2-c^2) (-SA SB-S (-2 S+Sqrt(3) SC)))+a^2 ((-S (-2 S+Sqrt(3) SB)-SA SC)^2-(-S (-2 S+Sqrt(3) SB)-SA SC) (-SA SB-S (-2 S+Sqrt(3) SC))+(-SA SB-S (-2 S+Sqrt(3) SC))^2) : :

X(14082) lies on the curve Q041 and these lines: {2, 5470}, {137, 629}, {636, 2381}

X(14082) = orthoptic-circle-of-Steiner-inellipe inverse of X(5983)
X(14082) = circumcircle-of-outer-Napoleon-triangle-inverse of X(14144)
X(14082) = psi-transform of X(628)

X(14083) =  SINGULAR FOCUS OF THE CUBIC K340

Barycentrics    a^2*(a^13 - a^12*b - 2*a^11*b^2 + 2*a^10*b^3 - a^9*b^4 + a^8*b^5 + 4*a^7*b^6 - 4*a^6*b^7 - a^5*b^8 + a^4*b^9 - 2*a^3*b^10 + 2*a^2*b^11 + a*b^12 - b^13 - a^12*c + a^11*b*c - a^9*b^3*c + 3*a^8*b^4*c - 2*a^7*b^5*c + 2*a^5*b^7*c - 3*a^4*b^8*c + a^3*b^9*c - a*b^11*c + b^12*c - 2*a^11*c^2 + 9*a^9*b^2*c^2 - 3*a^8*b^3*c^2 - 9*a^7*b^4*c^2 + 3*a^6*b^5*c^2 - 3*a^5*b^6*c^2 + 5*a^4*b^7*c^2 + 7*a^3*b^8*c^2 - 7*a^2*b^9*c^2 - 2*a*b^10*c^2 + 2*b^11*c^2 + 2*a^10*c^3 - a^9*b*c^3 - 3*a^8*b^2*c^3 + 9*a^7*b^3*c^3 - 3*a^6*b^4*c^3 - 6*a^5*b^5*c^3 + 7*a^4*b^6*c^3 - 5*a^3*b^7*c^3 - a^2*b^8*c^3 + 3*a*b^9*c^3 - 2*b^10*c^3 - a^9*c^4 + 3*a^8*b*c^4 - 9*a^7*b^2*c^4 - 3*a^6*b^3*c^4 + 18*a^5*b^4*c^4 - 8*a^4*b^5*c^4 - 7*a^3*b^6*c^4 + 5*a^2*b^7*c^4 + 3*a*b^8*c^4 - b^9*c^4 + a^8*c^5 - 2*a^7*b*c^5 + 3*a^6*b^2*c^5 - 6*a^5*b^3*c^5 - 8*a^4*b^4*c^5 + 12*a^3*b^5*c^5 + a^2*b^6*c^5 - 2*a*b^7*c^5 + b^8*c^5 + 4*a^7*c^6 - 3*a^5*b^2*c^6 + 7*a^4*b^3*c^6 - 7*a^3*b^4*c^6 + a^2*b^5*c^6 - 4*a*b^6*c^6 - 4*a^6*c^7 + 2*a^5*b*c^7 + 5*a^4*b^2*c^7 - 5*a^3*b^3*c^7 + 5*a^2*b^4*c^7 - 2*a*b^5*c^7 - a^5*c^8 - 3*a^4*b*c^8 + 7*a^3*b^2*c^8 - a^2*b^3*c^8 + 3*a*b^4*c^8 + b^5*c^8 + a^4*c^9 + a^3*b*c^9 - 7*a^2*b^2*c^9 + 3*a*b^3*c^9 - b^4*c^9 - 2*a^3*c^10 - 2*a*b^2*c^10 - 2*b^3*c^10 + 2*a^2*c^11 - a*b*c^11 + 2*b^2*c^11 + a*c^12 + b*c^12 - c^13) : :

X(14083) lies on these lines: {3, 14680}, {120, 4220}

X(14083) = circumcircle-inverse-of X(14680)

X(14084) =  SINGULAR FOCUS OF THE CUBIC K792

Barycentrics    a^2*(4*a^8*b^4 - 2*a^6*b^6 - 4*a^4*b^8 + 2*a^2*b^10 - 13*a^8*b^2*c^2 + 18*a^6*b^4*c^2 - 29*a^4*b^6*c^2 + 5*a^2*b^8*c^2 - b^10*c^2 + 4*a^8*c^4 + 18*a^6*b^2*c^4 + 9*a^4*b^4*c^4 + 12*a^2*b^6*c^4 - 2*b^8*c^4 - 2*a^6*c^6 - 29*a^4*b^2*c^6 + 12*a^2*b^4*c^6 - 2*b^6*c^6 - 4*a^4*c^8 + 5*a^2*b^2*c^8 - 2*b^4*c^8 + 2*a^2*c^10 - b^2*c^10) : :

X(14084) lies on these lines: {351, 14830}, {353, 9486}, {543, 14684}, {729, 2080}, {3398, 9145}, {5969, 7618}, {6234, 11654}, {11621, 14666}

X(14085) =  ISOGONAL CONJUGATE OF X(14078)

Barycentrics    a2 |(a - b)(a - c)| : :

X(14085) lies on this line: {14087,14089}

X(14085) = isogonal conjugate of X(14078)
X(14085) = anticomplement of the complementary conjugate of X(14083)
X(14085) = barycentric product X(i)*X(j) for these {i, j}: {6, 14087}, {42, 14089}, {765, 14079}, {1016, 14088}, {1110, 14080}, {1252, 14078}
X(14085) = barycentric quotient X(i)/X(j) for these (i, j): (1, 14080), (31, 14079), (32, 14088), (42, 14086), (1252, 14087), (1918, 14090)
X(14085) = trilinear product X(i)*X(j) for these {i, j}: {31, 14087}, {213, 14089}, {765, 14088}, {1110, 14078}, {1252, 14079}
X(14085) = trilinear quotient X(i)/X(j) for these (i, j): (2, 14080), (6, 14079), (31, 14088), (37, 14086), (213, 14090), (765, 14087)
X(14085) = X(i)-isoconjugate-of-X(j) for these {i,j}: {2, 14079}, {6, 14080}, {75, 14088}, {81, 14086}
X(14085) = X(i)-reciprocal conjugate of-X(j) for these (i,j): (1, 14080), (31, 14079), (32, 14088), (42, 14086)

X(14086) =  BSS(a → |b2 - c2|) of X(1)

Trilinears    |sin(B - C)| : :
Trilinears    |A'A"|, where A'B'C', A"B"C" are the medial and orthic triangles
Trilinears    f(A,B,C) : :, where f(A,B,C) is the length of the segment cut off from BC by the nine-point circle
Trilinears    sin A' : :, where A' is the angle formed by the internal tangents to the incircle and A-excircle
Trilinears    sin A" : :, where A" is the angle formed by the external tangents to the B- and C-excircles
Barycentrics    |b2 - c2| : :

If ABC is acute, then X(14086) is the vertex of the Schroeter triangle corresponding to the middle-angled vertex of ABC; if ABC is obtuse, then X(14086) is the vertex of the Schroeter triangle corresponding to the least-angled vertex of ABC. (Randy Hutson, November 6 2017)

X(14086) lies on the cubics K237, K238, K239, K672, K877, K878, K1144 and this line: {14078,14079}

X(14086) = isotomic conjugate of X(14089)
X(14086) = isotomic conjugate of the isogonal conjugate of X(14090)
X(14086) = crosspoint of X(14078) and X(14080
X(14086) = barycentric product X(i)*X(j) for these {i, j}: {10, 14078}, {37, 14080}, {76, 14090}, {115, 14089}, {313, 14088}, {321, 14079}
X(14086) = barycentric quotient X(i)/X(j) for these (i, j): (2, 14089), (10, 14087), (42, 14085), (3120, 14078), (3122, 14088), (3124, 14090), (3125, 14079), (14078, 86), (14079, 81), (14080, 274), (14085, 4570), (14087, 4600), (14088, 58), (14089, 4590), (14090, 6), (16732, 14080)
X(14086) = trilinear product X(i)*X(j) for these {i, j}: {{10, 14078}, {37, 14080}, {76, 14090}, {115, 14089}, {313, 14088}, {321, 14079}, {3120, 14087}, {14085, 21207}
X(14086) = trilinear quotient X(i)/X(j) for these (i, j): (37, 14085), (321, 14087), (1109, 14086)
X(14086) = X(i)-isoconjugate of X(j) for these (i,j): {{31, 14089}, {81, 14085}, {1101, 14086}, {1333, 14087}, {4567, 14088}, {4570, 14079}, {14090, 24041}
X(14086) = X(i)-reciprocal conjugate of-X(j) for these (i,j): (10, 14087), (42, 14085)
X(14086) = diagonal crosspoint of the Feuerbach quadrilateral

X(14087) =  BSS(a → |b - c|-1) of X(1)

Barycentrics    |(a - b)(a - c)| : :

X(14087) lies on these lines: {2,14084}, {1016,14083}, {14085,14089}

X(14087) = isogonal conjugate of X(14088)
X(14087) = isotomic conjugate of X(14078)
X(14087) = anticomplement of X(14084)
X(14087) = barycentric product X(i)*X(j) for these {i, j}: {10, 14089}, {76, 14085}, {765, 14080}, {1016, 14078}
X(14087) = barycentric quotient X(i)/X(j) for these (i, j): (1, 14079), (10, 14086), (42, 14090), (75, 14080), (1016, 14087), (1252, 14085)
X(14087) = trilinear product X(i)*X(j) for these {i, j}: {37, 14089}, {75, 14085}, {765, 14078}, {1016, 14079}, {1252, 14080}
X(14087) = trilinear quotient X(i)/X(j) for these (i, j): (2, 14079), (37, 14090), (76, 14080), (321, 14086), (765, 14085)
X(14087) = X(i)-isoconjugate-of-X(j) for these {i,j}: {6, 14079}, {32, 14080}, {81, 14090}, {244, 14085}
X(14087) = X(i)-reciprocal conjugate of-X(j) for these (i,j): (1, 14079), (10, 14086), (42, 14090), (75, 14080)

X(14088) =  ISOGONAL CONJUGATE OF X(14087)

Barycentrics    a2|b - c| : :

X(14088) lies on these lines: {}

X(14088) = isogonal conjugate of X(14087)
X(14088) = anticomplement of the complementary conjugate of X(14084)
X(14088) = barycentric product X(i)*X(j) for these {i, j}: {1, 14079}, {6, 14078}, {31, 14080}, {58, 14086}, {86, 14090}, {1015, 14087}
X(14088) = barycentric quotient X(i)/X(j) for these (i, j): (32, 14085), (58, 14089), (244, 14080), (1015, 14078), (1977, 14088)
X(14088) = trilinear product X(i)*X(j) for these {i, j}: {6, 14079}, {31, 14078}, {32, 14080}, {81, 14090}, {244, 14085}, {1333, 14086}
X(14088) = trilinear quotient X(i)/X(j) for these (i, j): (31, 14085), (81, 14089), (244, 14078), (1015, 14079), (1086, 14080)
X(14088) = X(i)-isoconjugate-of-X(j) for these {i,j}: {37, 14089}, {75, 14085}, {765, 14078}, {1016, 14079}
X(14088) = X(i)-reciprocal conjugate of-X(j) for these (i,j): (32, 14085), (58, 14089), (244, 14080), (1015, 14078)

X(14089) =  BSS(a → |b2 - c2|-1) of X(1)

Barycentrics    |(a2 - b2)(a2 - c2)| : :

X(14089) lies on this line: {14085,14087}

X(14089) = isogonal conjugate of X(14090)
X(14089) = isotomic conjugate of X(14086)
X(14089) = barycentric product X(i)*X(j) for these {i, j}: {86, 14087}, {310, 14085}
X(14089) = barycentric quotient X(i)/X(j) for these (i, j): (58, 14088), (81, 14079), (86, 14078), (274, 14080)
X(14089) = trilinear product X(i)*X(j) for these {i, j}: {81, 14087}, {274, 14085}
X(14089) = trilinear quotient X(i)/X(j) for these (i, j): (81, 14088), (86, 14079), (274, 14078), (310, 14080)
X(14089) = X(i)-isoconjugate-of-X(j) for these {i,j}: {37, 14088}, {42, 14079}, {213, 14078}
X(14089) = X(i)-reciprocal conjugate of-X(j) for these (i,j): (58, 14088), (81, 14079), (86, 14078), (274, 14080)

X(14089) = isogonal conjugate of X(14090)
X(14089) = isotomic conjugate of X(14086)

X(14090) =  ISOGONAL CONJUGATE OF X(14089)

Barycentrics    a2|b2 - c2| : :

X(14090) lies on the cubic K554 and these lines: {6685,14083}, {14078,14079}

X(14090) = isogonal conjugate of X(14089)
X(14090) = barycentric product X(i)*X(j) for these {i, j}: {6, 14086}, {10, 14088}, {37, 14079}, {42, 14078}, {213, 14080}
X(14090) = barycentric quotient X(i)/X(j) for these (i, j): (42, 14087), (1084, 14090), (1918, 14085)
X(14090) = trilinear product X(i)*X(j) for these {i, j}: {31, 14086}, {37, 14088}, {42, 14079}, {213, 14078}, {1918, 14080}
X(14090) = trilinear quotient X(i)/X(j) for these (i, j): (37, 14087), (213, 14085)
X(14090) = X(i)-isoconjugate-of-X(j) for these {i,j}: {81, 14087}, {274, 14085}
X(14090) = X(42)-reciprocal conjugate of-X(14087)

X(14091) =  X(2)-CEVA CONJUGATE OF X(235)

Barycentrics    a^2 (a^2+b^2-c^2) (a^2-b^2+c^2) (a^4 b^2-2 a^2 b^4+b^6+a^4 c^2+4 a^2 b^2 c^2-b^4 c^2-2 a^2 c^4-b^2 c^4+c^6) (a^8-2 a^6 b^2+2 a^2 b^6-b^8-2 a^6 c^2+10 a^4 b^2 c^2-6 a^2 b^4 c^2-2 b^6 c^2-6 a^2 b^2 c^4+6 b^4 c^4+2 a^2 c^6-2 b^2 c^6-c^8) : :

X(14091) lies on the cubic K924 and these lines: {6,64}, {235,800}, {6753,14401}

X(14091) = X(i)-complementary conjugate of X(j) for these (i,j): {31, 235}, {1660, 10}, {11413, 2887}
X(14091) = X(2)-Ceva conjugate of X(235)
X(14091) = crosspoint of X(2) and X(11413)
X(14091) = barycentric product X(235)*X(11413)

X(14092) =  X(2)-CEVA CONJUGATE OF X(64)

Barycentrics    a^2 (a^4-2 a^2 b^2+b^4+2 a^2 c^2+2 b^2 c^2-3 c^4) (a^4+2 a^2 b^2-3 b^4-2 a^2 c^2+2 b^2 c^2+c^4) (a^10+3 a^8 b^2-14 a^6 b^4+14 a^4 b^6-3 a^2 b^8-b^10+3 a^8 c^2+20 a^6 b^2 c^2-14 a^4 b^4 c^2-12 a^2 b^6 c^2+3 b^8 c^2-14 a^6 c^4-14 a^4 b^2 c^4+30 a^2 b^4 c^4-2 b^6 c^4+14 a^4 c^6-12 a^2 b^2 c^6-2 b^4 c^6-3 a^2 c^8+3 b^2 c^8-c^10) : :

X(14092) lies on the cubic K924 and these lines: {3,14390}, {6,14379}, {64,800}, {235,393}, {1073,13567}, {3343,6509}, {8573,14642}

X(14092) = X(i)-complementary conjugate of X(j) for these (i,j): {31, 64}, {1661, 10}, {6225, 2887}
X(14092) = X(2)-Ceva conjugate of X(64)
X(14092) = crosspoint of X(2) and X(6225)
X(14092) = barycentric product X(i)*X(j) for these {i,j}: {64, 6225}, {253, 1661}
X(14092) = barycentric quotient X(i)/X(j) for these {i,j}: {1661, 20}, {6225, 14615}

X(14093) =  MIDPOINT OF X(1656) AND X(3534)

Barycentrics    19*a^4-17*(b^2+c^2)*a^2-2*(b^ 2-c^2)^2 : :
X(14093) = 2*X(2)-7*X(3) = 17*X(2)-7*X(4) = 19*X(2)-14*X(5) = 13*X(2)+7*X(20) = 3*X(2)+7*X(376) = 12*X(2)-7*X(381) = X(2)+4*X(548) = 9*X(2)-14*X(549) = 14*X(3098)+X(6144) = 14*X(3579)+X(3633)

As a point on the Euler line, X(14093) has Shinagawa coefficients (17, -21).

See Antreas Hatzipolakis and César Lozada, Hyacinthos 26495.

X(14093) lies on these lines: {2, 3}, {3098, 6144}, {3579, 3633}, {3625, 3654}, {3635, 12702}, {3656, 12512}, {5023, 5346}, {5210, 5309}, {6446, 9541}, {6448, 9681}, {6496, 13903}, {6497, 13961}, {11592, 12290}, {11694, 12244}, {11850, 11999}

X(14093) = midpoint of X(1656) and X(3534)
X(14093) = reflection of X(i) in X(j) for these (i,j): (632, 12100), (3522, 8703), (3830, 3091), (3843, 2), (5071, 549)
X(14093) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2,3,15706), (2, 3850, 5055), (3, 3830, 3524), (3, 5073, 3530), (376, 381, 3534), (381, 3526, 547), (547, 3830, 381), (631, 5076, 1656), (1656, 3843, 5072), (3091, 3526, 1656), (3135, 14065, 7462), (3524, 3830, 3526), (3839, 10299, 11812), (3839, 11812, 5070)

X(14094) =  REFLECTION OF X(74) IN X(110)

Barycentrics    a^2 (a^8-5 a^6 b^2+9 a^4 b^4-7 a^2 b^6+2) : :
X(14094) = 4 X(3) - 3 X(74) = 2 X(3) - 3 X(110) = X(74) - 4 X(399) = X(3) - 3 X(399) = 3 X(265) - 4 X(546) = 4 X(576) - 3 X(895) = 5 X(74) - 8 X(1511) = 5 X(3) - 6 X(1511) = 5 X(110) - 4 X(1511) = 5 X(399) - 2 X(1511) = 6 X(125) - 7 X(3090) = 6 X(113) - 5 X(3091) = 3 X(146) - X(3146) = 5 X(3091) - 3 X(3448) = 6 X(1539) - 5 X(5076) = 3 X(74) - 8 X(5609) = 3 X(1511) - 5 X(5609) = 3 X(110) - 4 X(5609)

X(14094) is the point denoted by F, described at Cubics: K913

Let A'B'C' be the antipode-in-anticomplementary-circle of the anticomplementary triangle. Let L, M, N be lines through A', B', C', respectively, parallel to the Euler line. Let L' be the reflection of L in sideline BC, let M' be the reflection of M in sideline CA, and let N' be the reflection of N in sideline AB. The lines L', M', N' concur in X(14094). (Randy Hutson, November 2, 2017)

X(14094) lies on the cubic K854 and these lines: {3,74}, {4,542}, {5,5643}, {20,541}, {25,13148}, {54,7527}, {61,10657}, {62,10658}, {113,3091}, {125,3090}, {146,3146}, {155,12290}, {265,546}, {376,10990}, {511,12112}, {547,13393}, {575,11579}, {631,5642}, {632,10272}, {1112,5198}, {1181,5622}, {1498,2781}, {1503,10510}, {1533,5965}, {1539,5076}, {1986,10594}, {1993,11455}, {1995,5890}, {2771,7984}, {2777,3529}, {2854,10752}, {2948,7991}, {3024,3303}, {3028,3304}, {3089,12828}, {3284,13509}, {3292,6000}, {3518,7722}, {3520,9705}, {3525,5972}, {3567,12824}, {3581,12105}, {3627,7728}, {3628,10264}, {3746,7727}, {3857,11801}, {4550,11003}, {5066,12834}, {5504,12086}, {5562,8718}, {5563,10091}, {5621,7509}, {5656,9968}, {5889,7530}, {5891,7496}, {5907,7550}, {6102,7545}, {6419,12375}, {6420,12376}, {6453,10819}, {6454,10820}, {6519,10817}, {6522,10818}, {6699,10303}, {6759,7556}, {7555,7691}, {7575,10540}, {7724,12381}, {7978,7982}, {8542,11180}, {9138,11615}, {9716,13352}, {9934,10628}, {11403,12133}, {11439,12161}, {12164,12271}, {13482,13596}

X(14094) = midpoint of X(399) and X(12308)
X(14094) = reflection of X(i) in X(j) for these {i,j}: {3, 5609}, {74, 110}, {110, 399}, {125, 6053}, {895, 9970}, {3448, 113}, {7464, 3292}, {9140, 5655}, {10620, 1511}, {10721, 146}, {10733, 7728}, {12281, 12825}, {12284, 1986}, {12317, 125}, {12902, 1539}
X(14094) = anticomplement of X(16003)
X(14094) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 399, 5609), (3, 5609, 110)

X(14095) =  (name pending)

Barycentrics    2 a^22-11 a^20 b^2+20 a^18 b^4-a^16 b^6-48 a^14 b^8+70 a^12 b^10-28 a^10 b^12-30 a^8 b^14+46 a^6 b^16-27 a^4 b^18+8 a^2 b^20-b^22-11 a^20 c^2+50 a^18 b^2 c^2-98 a^16 b^4 c^2+137 a^14 b^6 c^2-185 a^12 b^8 c^2+179 a^10 b^10 c^2-45 a^8 b^12 c^2-97 a^6 b^14 c^2+108 a^4 b^16 c^2-45 a^2 b^18 c^2+7 b^20 c^2+20 a^18 c^4-98 a^16 b^2 c^4+212 a^14 b^4 c^4-246 a^12 b^6 c^4+104 a^10 b^8 c^4+47 a^8 b^10 c^4+34 a^6 b^12 c^4-162 a^4 b^14 c^4+110 a^2 b^16 c^4-21 b^18 c^4-a^16 c^6+137 a^14 b^2 c^6-246 a^12 b^4 c^6+96 a^10 b^6 c^6+19 a^8 b^8 c^6+12 a^6 b^10 c^6+108 a^4 b^12 c^6-160 a^2 b^14 c^6+35 b^16 c^6-48 a^14 c^8-185 a^12 b^2 c^8+104 a^10 b^4 c^8+19 a^8 b^6 c^8+10 a^6 b^8 c^8-27 a^4 b^10 c^8+170 a^2 b^12 c^8-34 b^14 c^8+70 a^12 c^10+179 a^10 b^2 c^10+47 a^8 b^4 c^10+12 a^6 b^6 c^10-27 a^4 b^8 c^10-166 a^2 b^10 c^10+14 b^12 c^10-28 a^10 c^12-45 a^8 b^2 c^12+34 a^6 b^4 c^12+108 a^4 b^6 c^12+170 a^2 b^8 c^12+14 b^10 c^12-30 a^8 c^14-97 a^6 b^2 c^14-162 a^4 b^4 c^14-160 a^2 b^6 c^14-34 b^8 c^14+46 a^6 c^16+108 a^4 b^2 c^16+110 a^2 b^4 c^16+35 b^6 c^16-27 a^4 c^18-45 a^2 b^2 c^18-21 b^4 c^18+8 a^2 c^20+7 b^2 c^20-c^22 : :

See Antreas Hatzipolakis and César Lozada, Hyacinthos 26442.

X(14095) lies on this line: {10205,15345}

X(14096) =  MONTESDEOCA-EULER POINT

Barycentrics    a^2 (b^2+c^2) (-a^4+2 b^2 c^2+a^2 (b^2+c^2)) : :

See Angel Montesdeoca, Hyacinthos 26454.

X(14096) lies on these lines: {2, 3}, {6, 11175}, {32, 10014}, {39, 3051}, {51, 5188}, {95, 6394}, {141, 1634}, {160, 3763}, {216, 9475}, {263, 1350}, {574, 3117}, {592, 3398}, {597, 5201}, {669, 11183}, {1078, 3978}, {1613, 5013}, {1899, 7800}, {2076, 8570}, {2979, 3095}, {3060, 9821}, {3589, 8266}, {3819, 13334}, {5012, 12054}, {5024, 9463}, {5092, 5191}, {5106, 8589}, {5650, 9155}, {7767, 11245}, {7998, 11171}

X(14096) = crossdifference of every pair of points on line X(647)X(4108)

X(14097) =  (name pending)

Trilinears    (2*cos(2*A)-4*cos(4*A)+12)* cos(B-C)+(8*cos(A)-6*cos(3*A)) *cos(2*(B-C))+(2*cos(2*A)-2* cos(4*A)+3)*cos(3*(B-C))+2* cos(A)*cos(4*(B-C))+cos(7*A)+ 5*cos(A)-3*cos(3*A)-2*cos(5*A) : :

See Tran Quang Hung and César Lozada, Hyacinthos 26471.

X(14097) lies on these lines: {5, 252}, {110, 10203}

X(14097) = circumcevian isogonal conjugate of X(5)

X(14098) =  (name pending)

Trilinears    (205*cos(2*A)+254*cos(4*A)- 103*cos(6*A)-5*cos(8*A)+3*cos( 10*A)-665/2)*cos(B-C)+(-295* cos(A)+323*cos(3*A)-23*cos(5* A)-41*cos(7*A)+7*cos(9*A))* cos(2*(B-C))+(85*cos(2*A)+121* cos(4*A)-53*cos(6*A)+4*cos(8* A)-158)*cos(3*(B-C))+(-88*cos( A)+88*cos(3*A)-10*cos(7*A))* cos(4*(B-C))+(5*cos(2*A)+19* cos(4*A)-7*cos(6*A)-25)*cos(5* (B-C))+(-8*cos(A)+7*cos(3*A)+ cos(5*A))*cos(6*(B-C))-209* cos(A)+233*cos(3*A)+7*cos(9*A) +cos(11*A)-13*cos(5*A)-33*cos( 7*A) : :

See Tran Quang Hung and César Lozada, Hyacinthos 26471.

X(14098) lies on this line: {5, 252}

X(14098) =

X(14099) =  (name pending)

Barycentrics    (-2*sqrt(3)*S-a^2+b^2+c^2)*(a^ 2-b^2+c^2)*(a^2+b^2-c^2)*((2* a^22-4*(b^2+c^2)*a^20-2*(2*b^ 4-b^2*c^2+2*c^4)*a^18+26*(b^2+ c^2)*b^2*c^2*a^16+2*(25*b^8+ 25*c^8-b^2*c^2*(27*b^4+19*b^2* c^2+27*c^4))*a^14-2*(b^2+c^2)* (40*b^8+40*c^8-b^2*c^2*(53*b^ 4-47*b^2*c^2+53*c^4))*a^12+2*( 9*b^12+9*c^12+(38*b^8+38*c^8- 11*b^2*c^2*(3*b^4-4*b^2*c^2+3* c^4))*b^2*c^2)*a^10+4*(b^4-c^ 4)*(b^2-c^2)*(11*b^8+11*c^8-b^ 2*c^2*(26*b^4-31*b^2*c^2+26*c^ 4))*a^8-2*(b^2-c^2)^2*(10*b^ 12+10*c^12-(9*b^8+9*c^8-7*b^2* c^2*(3*b^4+4*b^2*c^2+3*c^4))* b^2*c^2)*a^6-2*(b^4-c^4)*(b^2- c^2)^3*(2*b^4-3*b^2*c^2+2*c^4) *(5*b^4+3*b^2*c^2+5*c^4)*a^4+ 2*(b^2-c^2)^6*(9*b^8+9*c^8+b^ 2*c^2*(13*b^4+14*b^2*c^2+13*c^ 4))*a^2-2*(b^2-c^2)^8*(b^2+c^ 2)*(2*b^4+3*b^2*c^2+2*c^4))*S* sqrt(3)+3*a^24-15*(b^2+c^2)*a^ 22+3*(6*b^4+17*b^2*c^2+6*c^4)* a^20+6*(b^2+c^2)*(5*b^4-11*b^ 2*c^2+5*c^4)*a^18-3*(31*b^8+ 31*c^8+3*b^2*c^2*(6*b^4+b^2*c^ 2+6*c^4))*a^16+3*(b^2+c^2)*( 21*b^8+21*c^8+2*b^2*c^2*(7*b^ 4+5*b^2*c^2+7*c^4))*a^14+3*(7* b^12+7*c^12-(29*b^8+29*c^8+2* b^2*c^2*(11*b^4+16*b^2*c^2+11* c^4))*b^2*c^2)*a^12-3*(b^2+c^ 2)*(b^4+c^4)*(b^8+c^8+2*b^2*c^ 2*(b^4-12*b^2*c^2+c^4))*a^10- 3*(b^2-c^2)^2*(30*b^12+30*c^ 12-(17*b^8+17*c^8-b^2*c^2*(17* b^4+24*b^2*c^2+17*c^4))*b^2*c^ 2)*a^8+6*(b^4-c^4)*(b^2-c^2)*( 18*b^12+18*c^12-(43*b^8+43*c^ 8-2*b^2*c^2*(29*b^4-30*b^2*c^ 2+29*c^4))*b^2*c^2)*a^6-3*(b^ 2-c^2)^4*(17*b^12+17*c^12-6*b^ 2*c^2*(2*b^8+b^4*c^4+2*c^8))* a^4+9*(b^2-c^2)^6*(b^2+c^2)*( b^8+c^8-2*b^2*c^2*(b^4+c^4))* a^2+3*(b^2-c^2)^8*b^2*c^2*(b^ 2+c^2)^2) : :

See Tran Quang Hung and César Lozada, Hyacinthos 26474.

X(14099) lies on this line: {4,16}

X(14100) =  X(9)X(55)∩X(11)X(142)

Barycentrics    a*(-a+b+c)*((b+c)*a^2+(b-c)^2* (-2*a+b+c)) : :
X(14100) = 2*X(7)-3*X(354) = X(7)-3*X(7671) = 4*X(9)-3*X(210) = 3*X(210)-2*X(3059) = 4*X(2550)-5*X(3698) = 2*X(2951)-3*X(5918) = X(3962)-4*X(5698)

See Antreas Hatzipolakis and César Lozada, Hyacinthos 26475.

X(14100) lies on these lines: {1,971}, {4,12710}, {6,4319}, {7,354}, {9,55}, {10,9844}, {11,142}, {33,608}, {37,2293}, {40,10398}, {44,1253}, {56,5732}, {57,2951}, {65,516}, {72,4314}, {144,145}, {181,10443}, {224,1001}, {226,7965}, {241,1742}, {527,3058}, {528,12743}, {938,9943}, {942,4312}, {946,12671}, {954,1898}, {960,4313}, {1058,12675}, {1100,4336}, {1155,1445}, {1156,2346}, {1317,2801}, {1386,3100}, {1418,3000}, {1479,5805}, {1486,2182}, {1697,5223}, {1699,11018}, {1708,7964}, {1721,5228}, {1827,1839}, {1837,2550}, {1852,1890}, {2098,3243}, {3085,5817}, {3254,3255}, {3295,5779}, {3304,4321}, {3488,6001}, {3555,5850}, {3601,5696}, {3616,10861}, {3649,4890}, {3666,4335}, {3740,5281}, {3742,5274}, {3748,8545}, {3753,12690}, {3893,5853}, {4003,7004}, {5173,9580}, {5432,6666}, {5435,10178}, {5693,9957}, {5735,9670}, {5759,7957}, {5880,10940}, {5927,13405}, {6172,10385}, {6173,11238}, {7354,12573}, {8257,11502}, {9355,9440}, {10167,11019}, {10202,10738}, {10442,10473}, {10572,13375}, {10578,10865}, {11496,12664}

X(14100) = midpoint of X(390) and X(10394)
X(14100) = reflection of X(i) in X(j) for these (i,j): (7,5572), (65,5728), (354,7671), (3057,390), (3059,9), (4312,942), (5784,1001), (7354,12573), (7957,5759), (8581,1), (12688,11372)
X(14100) = crosspoint of X(7) and X(9)
X(14100) = crosssum of X(55) and X(57)
X(14100) = X(9)-of-Mandart-incircle-triangle
X(14100) = X(69)-of intouch-triangle
X(14100) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 9848, 10866), (1, 12680, 9850), (7, 5572, 354), (7, 7671, 5572), (9, 3059, 210), (9, 3174, 480), (9, 4326, 55), (55, 1864, 210), (480, 3174, 3689), (497, 10391, 354), (2293, 2310, 37), (8581, 10384, 10866)

X(14101) =  X(1)X(1263)∩X(11)X(137)

Barycentrics    (-a+b+c)*(a^6-(b^2+c^2)*a^4-( b^4+c^4+b*c*(2*b^2+b*c+2*c^2)) *a^2+(b^2-c^2)^2*(b+c)^2)*(a^ 3+(b+c)*a^2-(b^2-b*c+c^2)*a-( b^2-c^2)*(b-c))*(b-c)^2 : :
X(14101) = (R - 2r)*X(11) + 4r*X(137)

See Antreas Hatzipolakis and César Lozada, Hyacinthos 26481.

X(14101) lies on the incentral circle and these lines: {1, 1263}, {11, 137}, {36, 12026}, {55, 11671}, {128, 3614}, {498, 13512}, {930, 5432}, {1141, 7354}, {5326, 13372}, {7951, 14072}, {10592, 14073}

X(14101) = {X(137), X(3327)}-harmonic conjugate of X(11)

X(14102) =  X(35)X(500)∩X(36)X(6150)

Barycentrics    a^2*(a^2-b^2-b*c-c^2)*(a^6-2*( b^2+c^2)*a^4+(b^4+c^4+b*c*(b^ 2-b*c+c^2))*a^2-(b^2-c^2)^2*b* c)*(a^3+(b+c)*a^2-(b^2-b*c+c^ 2)*a-(b^2-c^2)*(b-c)) : :
X(14102) = R*X(1) + 2r*X(1157)

See Antreas Hatzipolakis and César Lozada, Hyacinthos 26481.

X(14102) lies on the incentral circle and these lines: {1, 1157}, {35, 500}, {36, 6150}

X(14103) =  COMPLEMENT OF X(12092)

Barycentrics    (3*SA^2-10*R^2*SA+3*S^2-4*SB* SC)*(SA^2+2*S^2+2*SB*SC-SW^2)* (SA+SW-5*R^2) : :

See Antreas Hatzipolakis and César Lozada, Hyacinthos 26484.

X(14103) lies on the nine-point circle and these lines: {2, 12092}, {113, 5876}

X(14103) = complement of X(12092)

X(14104) =  POLAR-CIRCLE-INVERSE OF X(5964)

Barycentrics    Sin(5 A)(2 Cos(B-C) (1+2 Cos(2 A)-4 Cos(A) Cos(B-C))+2 Cos(3 (B-C))+Sec(A)) : :

See Antreas Hatzipolakis and César Lozada, Hyacinthos 26484.

X(14104) lies on these lines: {4, 5964}, {6102, 6240}

X(14104) = polar circle-inverse-of-X(5964)

X(14105) =  (name pending)

Barycentrics    9*(12*S^2+23*SW^2)*SA^2-3*(39* S^2+59*SW^2)*SW*SA+9*(4*S^2+5* SW^2)*S^2-10*SW^4 : :

See Antreas Hatzipolakis and César Lozada, Hyacinthos 26495.

X(14105) lies on this line: {548, 7759}

X(14106) =  COMPLEMENT OF X(13150)

Barycentrics    SB*SC*((5*R^2-2*SW)*SA+3*(2*R^ 2-SW)*R^2+2*S^2) : :

See Tran Quang Hung and César Lozada, Hyacinthos 26497.

X(14106) lies on these lines: {2, 3}, {6750, 12300}

X(14106) = complement of X(13150)
X(14106) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1884, 7546, 444), (1884, 13852, 4238), (2675, 7557, 6622), (3147, 13362, 7401), (6906, 7557, 13383), (8368, 11099, 14035)

X(14107) =  X(124)X(125)∩X(1364)X(1365)

Barycentrics    a*((b+c)*a^8-b*c*a^7-(b+c)*(3* b^2-b*c+3*c^2)*a^6+b*c*(b^2-b* c+c^2)*a^5+(b+c)*(3*b^4+3*c^4- b*c*(b^2-7*b*c+c^2))*a^4+b*c*( b^4-4*b^2*c^2+c^4)*a^3-(b+c)*( b^6+c^6+(b^4+c^4+3*b*c*(b-c)^ 2)*b*c)*a^2-(b^4+c^4-3*b*c*(b- c)^2)*(b+c)^2*b*c*a+(b^3+c^3)* (b^2-c^2)^2*b*c)*(b-c)^2 : :

See Le Viet An and César Lozada, Hyacinthos 26501.

X(14107) lies on these lines: {124, 125}, {1364, 1365}

X(14108) =  X(125)X(2776)∩X(1086)X(1357)

Barycentrics    a*(b-c)^2*((b+c)*a^5+(b^2-b*c+ c^2)*a^4-(b^2-c^2)*(b-c)*a^3-( b^4+b^2*c^2+c^4)*a^2-2*b*c*(b+ c)*(b^2-4*b*c+c^2)*a+b*c*(b^2- 3*b*c+c^2)*(b+c)^2) : :

See Le Viet An and César Lozada, Hyacinthos 26501.

X(14108) lies on these lines: {125, 2776}, {1086, 1357}, {2802, 3178}

X(14109) =  X(125)X(2775)∩X(1358)X(1365)

Barycentrics    (b-c)^2*(a^5+(b+c)*a^4-(b+2*c) *(2*b+c)*a^3+2*(b+c)^3*a^2-(3* b^4+3*c^4+b*c*(b-c)^2)*a+(b^4- c^4)*(b-c)) : :

See Le Viet An and César Lozada, Hyacinthos 26501.

X(14109) lies on these lines: {125, 2775}, {512, 4904}, {1358, 1365}

X(14110) = REFLECTION OF X(65) IN X(3)

Trilinears    (b+c)*a^5-(b^2+6*b*c+c^2)*a^4-2*(b^2-c^2)*(b-c)*a^3+2*(b^4+c^4+2*b*c*(b^2-b*c+c^2))*a^2+(b^2-c^2)*(b-c)^3*a-(b^2-c^2)^2*(b-c)^2 : :
X(14110) = 3*X(165)-X(5903) = 3*X(210)-2*X(355) = 3*X(354)-4*X(1385) = 9*X(354)-8*X(6583) = 3*X(392)-2*X(946) = 5*X(631)-4*X(3812) = 2*X(942)-3*X(3576) = X(962)-3*X(3877) = 3*X(1385)-2*X(6583) = 2*X(1482)-3*X(5919)

Let ΛA be the circle through the orthogonal projections of the incenter and the A-excenter on AB and AC and define ΛB and ΛC cyclically. The bisecting circle of ΛA, ΛB and ΛC has squared-radius 2*(2*R^3-2*R*r^2-r^3)/R and center X(14110). (César Eliud Lozada, Aug. 11, 2017)

X(14110) lies on these lines: {1,3}, {2,7686}, {4,960}, {8,3427}, {10,6831}, {20,3869}, {30,5887}, {63,12114}, {72,515}, {78,11500}, {84,12526}, {102,3430}, {110,1295}, {210,355}, {219,1766}, {224,11682}, {255,1455}, {329,12667}, {376,9943}, {377,962}, {388,5758}, {392,442}, {411,4511}, {495,5763}, {516,3878}, {518,944}, {550,5918}, {573,2262}, {602,1104}, {631,3812}, {912,3962}, {970,5721}, {971,5693}, {997,3149}, {1012,12514}, {1108,2245}, {1290,2745}, {1329,1512}, {1350,3827}, {1478,5812}, {1737,6922}, {1753,1902}, {1765,4047}, {1788,6926}, {1836,6850}, {1837,6827}, {1858,6868}, {1864,10572}, {1898,7491}, {1905,7412}, {2704,2724}, {2771,12119}, {2800,10609}, {2801,4067}, {2817,5930}, {3485,6908}, {3486,6987}, {3528,10178}, {3555,5882}, {3556,11414}, {3560,3683}, {3616,13374}, {3698,6862}, {3740,5818}, {3753,6684}, {3754,10164}, {3838,6937}, {3868,5731}, {3880,6899}, {3884,4301}, {3916,5450}, {3983,5790}, {4018,5884}, {4292,12709}, {4295,6916}, {4304,12711}, {4305,10391}, {4313,12710}, {4640,6906}, {4679,6893}, {4847,10914}, {5044,5587}, {5086,6840}, {5087,6941}, {5250,11496}, {5439,10165}, {5440,6796}, {5603,6889}, {5657,5836}, {5691,5692}, {5730,6261}, {5762,8581}, {5806,8227}, {5880,6897}, {6361,6934}, {6825,11375}, {6860,9780}, {6907,12047}, {6917,12699}, {6925,11415}, {7971,12565}, {10179,10595}

X(14110) = midpoint of X(i) and X(j) for these {i,j}: {20, 3869}, {3057, 7957}, {3962, 12680}, {5697, 7991}
X(14110) = reflection of X(i) in X(j) for these (i,j): (4, 960), (65, 3), (3555, 5882), (3868, 12675), (4018, 5884), (4301, 3884), (5691, 5777), (7982, 9957), (10914, 11362), (11531, 13600), (12672, 3878), (12688, 5887)
X(14110) = anticomplement of X(7686)
X(14110) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 40, 3428), (1, 5903, 5173), (1, 7742, 1319), (3, 10306, 11507), (40, 2077, 3579), (40, 6282, 10310), (40, 10310, 13528), (165, 3612, 3), (3612, 5903, 13750), (5902, 7987, 9940), (5903, 13750, 65), (6244, 12702, 40)

X(14111) =  X(4)X(93)∩X(50)X(252)

Trilinears    csc A tan 3A - sec 3A : :
Trilinears    cos 2A sec 3A : :
Barycentrics    (S^2-SA^2)*(3*S^2-SB^2)*(3*S^ 2-SC^2)*SB*SC : :

See Antreas Hatzipolakis and César Lozada, Hyacinthos 26506.

X(14111) is the perspector of the Montesdeoca conic; see X(2904) and HG121017.

X(14111) lies on these lines: {4,93}, {50,252}, {54,14106}, {930,1299}, {1986,6801}, {2904,8745}

X(14111) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4, 562, 93), (93, 562, 3519)
X(14111) = isoconjugate of X(j) and X(j) for these (i,j): {49, 91}, {68, 2964}, {1820, 1994}
X(14111) = barycentric product X(i)*X(j) for these {i,j}: {24, 11140}, {93, 1993}, {252, 467}, {317, 2963}, {1748, 2962}, {3519, 11547}
X(14111) = barycentric quotient X(i)/X(j) for these {i,j}: {24, 1994}, {93, 5392}, {317, 7769}, {571, 49}, {2963, 68}, {6753, 1510}, {8745, 3518}

X(14112) =  MIDPOINT OF X(3756) AND X(5516)

Barycentrics    (3*a-b-c)*(a^3-4*(b+c)*a^2-(3* b^2-19*b*c+3*c^2)*a+(b+c)*(2* b^2-7*b*c+2*c^2))*(b-c)^2 : :
X(14112) = 16*X(3633)+X(5205) = 7*X(3756)+10*X(5510)

See Le Viet An and César Lozada, Hyacinthos 26508.

X(14112) lies on these lines: {1, 2}, {3667, 3756}

X(14112) = midpoint of X(3756) and X(5516)

X(14113) =  MIDPOINT OF X(2698) AND X(13137)

Barycentrics    a^2*(b^2-c^2)^2*(a^8-3*(b^2+c^ 2)*a^6+(3*b^4+b^2*c^2+3*c^4)* a^4-(b^4-c^4)*(b^2-c^2)*a^2-b^ 2*c^2*(b^4+c^4)) : :

See Le Viet An and César Lozada, Hyacinthos 26508.

X(14113) lies on these lines: {2, 12833}, {3, 6}, {115, 512}, {2698, 13137}, {3124, 3288}, {5663, 5915}, {6785, 7737}

X(14113) = midpoint of X(2698) and X(13137)
X(14113) = complement of X(12833)
X(14113) = orthogonal projection of X(115) on the Brocard axis
X(14113) = {X(2024), X(4289)}-harmonic conjugate of X(8554)

X(14114) =  X(6)X(13)∩X(690)(2682)

Barycentrics    (b^2-c^2)^2*(2*a^2-b^2-c^2)*( 4*a^14-7*(b^2+c^2)*a^12+10*b^ 2*c^2*a^10+2*(b^2+c^2)*(3*b^4- b^2*c^2+3*c^4)*a^8-2*(b^4+c^4) *(3*b^4+4*b^2*c^2+3*c^4)*a^6+( b^6+c^6)*(3*b^4+7*b^2*c^2+3*c^ 4)*a^4+2*(b^12+c^12-(7*b^8+7* c^8-b^2*c^2*(8*b^4-7*b^2*c^2+ 8*c^4))*b^2*c^2)*a^2+(b^4-c^4) *(b^2-c^2)*(-2*b^8-2*c^8+b^2* c^2*(6*b^4-5*b^2*c^2+6*c^4))) : :

See Le Viet An and César Lozada, Hyacinthos 26508.

X(14114) lies on these lines: {6, 13}, {690, 2682}

X(14115) =  X(1)X(3)∩X(11)(513)

Barycentrics    a*(a^5-(2*b^2-3*b*c+2*c^2)*a^ 3-b*c*(b+c)*a^2+(b^4+c^4-3*b* c*(b-c)^2)*a+(b^2-c^2)*(b-c)* b*c)*(b-c)^2*(-a+b+c) : :

See Le Viet An and César Lozada, Hyacinthos 26508.

X(14115) lies on these lines: {1, 3}, {11, 513}, {244, 1459}, {1364, 14027}, {3328, 5577}, {6073, 10956}

X(14115) = midpoint of X(i) and X(j) for these {i,j}: {11, 3025}, {3259, 3937}
X(14115) = complement of X(34151)
X(14115) = orthogonal projection of X(11) on line X(1)X(3)

X(14116) =  X(1)X(7)∩X(116)(514)

Barycentrics    (3*a^6-3*(b+c)*a^5-(4*b^2-3*b* c+4*c^2)*a^4+2*(b+c)^3*a^3+(3* b^4+3*c^4-2*b*c*(3*b^2+b*c+3* c^2))*a^2+(b^2-c^2)*(b-c)^3*a- (2*b^4+2*c^4+b*c*(b+c)^2)*(b- c)^2)*(b-c)^2 : :

See Le Viet An and César Lozada, Hyacinthos 26508.

X(14116) lies on these lines: {1, 7}, {116, 514}

X(14117) =  X(4)X(6)∩X(127)(525)

Barycentrics    (b^2-c^2)^2*(-a^2+b^2+c^2)*(5* a^14-6*(b^2+c^2)*a^12-(3*b^4- 7*b^2*c^2+3*c^4)*a^10+(b^2+c^ 2)*(2*b^4+3*b^2*c^2+2*c^4)*a^ 8-(b^4+c^4)*(b^2+c^2)^2*a^6+2* (b^4-c^4)*(b^2-c^2)*(3*b^4+b^ 2*c^2+3*c^4)*a^4-(b^6-c^6)*(b^ 2-c^2)*(b^4+6*b^2*c^2+c^4)*a^ 2-(b^4-c^4)*(b^2-c^2)^3*(2*b^ 4+b^2*c^2+2*c^4)) : :

See Le Viet An and César Lozada, Hyacinthos 26508.

X(14117) lies on these lines: {4, 6}, {127, 525}

X(14118) =  (name pending)

Barycentrics    a^2*(a^8 - 2*a^6*b^2 + 2*a^2*b^6 - b^8 - 2*a^6*c^2 + 3*a^4*b^2*c^2 - b^6*c^2 + 4*b^4*c^4 + 2*a^2*c^6 - b^2*c^6 - c^8) : :

X(14118) = isogonal conjugate of orthocenter of cevian triangle of X(264).

See Art of Problem Solving and Navneel Singhai and Bernard Gibert, Advanced Plane Geometry 3865.

X(14118) lies on these lines: {2, 3}, {32, 26216}, {35, 24025}, {49, 5876}, {52, 15033}, {54, 13754}, {64, 3796}, {74, 11562}, {99, 26166}, {110, 5907}, {143, 3581}, {155, 9545}, {156, 18435}, {182, 1204}, {184, 12111}, {185, 5012}, {265, 34826}, {323, 5562}, {388, 9672}, {389, 13434}, {390, 10831}, {497, 9659}, {511, 7691}, {542, 10619}, {567, 1199}, {569, 5890}, {578, 1994}, {962, 15177}, {1092, 11444}, {1147, 11459}, {1154, 37472}, {1176, 34146}, {1181, 11003}, {1192, 10601}, {1209, 17702}, {1498, 6800}, {1503, 32345}, {1511, 14128}, {1614, 12162}, {1619, 11469}, {1620, 17825}, {1968, 22240}, {1970, 22416}, {1993, 11425}, {2777, 32364}, {2883, 13394}, {2979, 13346}, {3043, 7723}, {3047, 12825}, {3060, 11424}, {3357, 10984}, {3410, 14516}, {3431, 15068}, {3448, 6146}, {3532, 10541}, {3564, 32333}, {3574, 35240}, {3580, 12241}, {3600, 10832}, {3619, 27082}, {3964, 32840}, {4550, 10539}, {5085, 8567}, {5422, 9786}, {5447, 10564}, {5462, 32110}, {5621, 11061}, {5656, 32321}, {5663, 10610}, {5888, 22978}, {5891, 12038}, {5893, 10117}, {5944, 10540}, {5961, 18300}, {6000, 15062}, {6101, 37495}, {6288, 30522}, {6759, 15305}, {7578, 13599}, {7592, 12163}, {7722, 12228}, {8193, 20070}, {8718, 14915}, {9306, 11449}, {9544, 11441}, {9590, 19925}, {9625, 18483}, {9626, 31673}, {9707, 18451}, {9729, 21663}, {9778, 37557}, {9970, 15021}, {10170, 12901}, {10282, 15030}, {10516, 35228}, {10545, 27355}, {10605, 37476}, {10627, 37477}, {10733, 22109}, {10982, 11002}, {11004, 12160}, {11204, 20791}, {11255, 18438}, {11412, 13352}, {11417, 11474}, {11418, 11473}, {11420, 11476}, {11421, 11475}, {11438, 15018}, {11439, 26881}, {11442, 19467}, {11468, 13336}, {11561, 12041}, {11591, 22115}, {12022, 12359}, {12054, 14675}, {12168, 14683}, {12203, 26214}, {12219, 15463}, {12220, 19124}, {12233, 14389}, {12279, 15080}, {12289, 18474}, {12294, 19121}, {12893, 14644}, {13293, 15055}, {13353, 13630}, {13445, 22352}, {13496, 38718}, {13568, 37649}, {13598, 15107}, {14627, 32608}, {14674, 25150}, {14805, 15032}, {14831, 37505}, {15034, 34802}, {15038, 16881}, {15060, 18350}, {15066, 35602}, {15740, 34438}, {16003, 18128}, {17821, 35264}, {18911, 26937}, {21243, 21659}, {23253, 35776}, {23263, 35777}, {25564, 38727}, {29012, 32332}, {31656, 34418}, {32210, 37471}, {32620, 38942}, {33537, 35259}, {33884, 37497}

X(14118) = isogonal conjugate of X(31976)
X(14118) = complement of X(34007)

X(14119) =  ORTHOGONAL PROJECTION OF X(19) ON EULER LINE

Barycentrics    (a+b) (a+c) (a^2+b^2-c^2) (a^2-b^2+c^2) (2 a^5-2 a^4 b-a^2 b^3+b^5-2 a^4 c+2 a^3 b c+a^2 b^2 c-b^4 c+a^2 b c^2-a^2 c^3-b c^4+c^5) : :

X(14119) lies on these lines: {2,3}, {19,523}, {1304,2724}

X(14119) = circumcircle-inverse of X(37908)
X(14119) = polar-circle inverse of X(857)
X(14119) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (27, 4241, 4), (28, 4244, 25)

X(14120) =  ORTHOGONAL PROJECTION OF X(115) ON EULER LINE

Barycentrics    (b-c)^2 (b+c)^2 (3 a^6-2 a^4 b^2-3 a^2 b^4+2 b^6-2 a^4 c^2+5 a^2 b^2 c^2-b^4 c^2-3 a^2 c^4-b^2 c^4+2 c^6) : :
X(14120) = 3 X(403) - X(1513) = X(1551) - 3 X(3545) = X(23) + 3 X(14041) = X(691) - 5 X(14061)

X(14120) lies on the circle {{X(2),X(115),X(125),X(5465)}} and these lines: {2,3}, {115,523}, {125,1499}, {524,5465}, {691,14061}, {3258,5512}, {8599,12079}

X(14120) = midpoint of X(i) and X(j) for these {i,j}: {115, 5099}, {125, 2682}, {7426, 8352}
X(14120) = complement X(7472)
X(14120) = X(2770)-Ceva conjugate of X(523)
X(14120) = crossdifference of every pair of points on line {647, 5467}
X(14120) = ninepoint-circle-inverse of X(3143)
X(14120) = polar-circle-inverse of X(4235)
X(14120) = orthoptic-circle-of-Steiner-inellipse-inverse of X(7417)
X(14120) = {X(1312),X(1313)}-harmonic conjugate of X(3143)

X(14121) =  MIDPOINT OF X(176) AND X(10405)

Trilinears    1/(-1 + sin A + cos A) : :
Barycentrics    2 a^6-a^5 (b+c)-a^4 (3 b^2+2 b c+3 c^2)+2 a^3 (b^3+b^2 c+b c^2+c^3)-a (b-c)^2 (b+c)^3+(b-c)^2 (b+c)^4+4 (-a^4+a^3 (b+c)-a^2 (b+c)^2-a (b-c)^2 (b+c)+2 (b^2-c^2)^2) r s : :
Barycentrics    1/(1 + cot A - csc A ) : :     (Peter Moses, August 18, 2017)
Barycentrics    1/(Ra - s) : : , where Ra, Rb, Rc are the exradii     Randy Hutson, June 27, 2018)
X(14121) = S X(4) - 2(r + 4 R) (r + 2 R - s) X(9)

See Tran Quang Hung and Angel Montesdeoca, Advanced Plane Geometry 3878.

X(14121) lies on these lines: {1,1336}, {2,175}, {4,9}, {29,2066}, {37,13911}, {44,13973}, {63,6348}, {92,1585}, {176,10405}, {189,13389}, {219,1378}, {226,13459}, {278,3535}, {388,6204}, {497,7347}, {637,1944}, {1123,1785}, {1146,3070}, {1743,13936}, {1788,6203}, {1851,3127}, {3061,7594}, {3305,6347}, {3812,13359}, {4194,7133}, {4662,13360}, {5393,13893}

X(14121) = midpoint X(176) and X(10405)
X(14121) = isogonal conjugate of X(2067)
X(14121) = complement X(175)
X(14121) = polar conjugate of X(1659)
X(14121) = X(318)-Ceva conjugate of X(7090)
X(14121) = cevapoint of X(1) and X(6212)
X(14121) = X(1)-cross conjugate of X(7090)
X(14121) = isoconjugate of X(j) and X(j) for these (i,j): {1, 2067}, {3, 2362}, {6, 13388}, {48, 1659}, {57, 5414}, {65, 1805}, {222, 7133}, {603, 7090}, {606, 13390}, {6213, 6502}
X(14121) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2,13386,13390), (4,281,7090), (9,10,7090), (92,1585,1659), (1861,7079,7090), (2345,2551,7090), (2550,6554,7090)
X(14121) = barycentric product X(i)*X(j) for these {i,j}: {8, 13390}, {264, 2066}, {318, 13389}, {6502, 7017}, {7090, 13386}
X(14121) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 13388}, {4, 1659}, {6, 2067}, {19, 2362}, {33, 7133}, {55, 5414}, {281, 7090}, {284, 1805}, {1336, 13390}, {1806, 1790}, {2066, 3}, {5414, 1335}, {6212, 13389}, {6502, 222}, {7090, 13387}, {7133, 6213}, {13389, 77}, {13390, 7}

X(14122) =  X(1)X(3)∩X(659)X(6164)

Barycentrics    (a (2 a-b-c) (a^2-a (b+c)-(b-c)^2+b c))/(b+c-a) : :

See Tran Quang Hung and Angel Montesdeoca, Advanced Plane Geometry 3880.

X(14122) lies on these lines: {1,3}, {659, 6164}, {3911,4759}

X(14123) =  (name pending)

Barycentrics    a^12-a^8 (b^4+b^2 c^2+c^4)+2 a^6 (b^6+c^6)+a^4 (b^8+b^6 c^2-2 b^4 c^4+b^2 c^6+c^8)-2 a^2 (b^10+b^6 c^4+b^4 c^6+c^10)-(b^4-c^4)^2 (b^4+c^4) : :

See Tran Quang Hung and Angel Montesdeoca, Advanced Plane Geometry 3884.

X(14123) lies on this line: {2,3}

X(14124) =  (name pending)

Trilinears    1/(a^4*(b^3+c^3)^2) : :

See Tran Quang Hung and César Lozada, Advanced Plane Geometry 3904.

X(14124) lies on these lines: {}

X(14124) = barycentric quotient X(983)/X(21815)
X(14124) = trilinear square of X(38810)
X(14124) = X(983)-reciprocal conjugate of-X(21815)

X(14125) =  (name pending)

Trilinears    (b^2+c^2)^3/a^3 : :

See Tran Quang Hung and César Lozada, Advanced Plane Geometry 3904.

X(14125) lies on these lines: {305,7855}, {3266,5041}

X(14125) = trilinear cube of X(141)

X(14126) =  X(2)X(6)∩X(111)X(9810)

Trilinears    a^6-2*(b+c)*a^5-(b^2-6*b*c+c^2)*a^4-(b+c)*(b^2-3*b*c+c^2)*a^3-(2*b^4+2*c^4-b*c*(3*b^2-7*b*c+3*c^2))*a^2+(b+c)*(b^4+c^4+b*c*(3*b^2-7*b*c+3*c^2))*a-3*b*c*(b^2+c^2)*(b-c)^2 : :

See Tran Quang Hung and César Lozada, Advanced Plane Geometry 3912.

X(14126) lies on these lines: {2,6}, {111,9810}

X(14127) =  X(2)X(3)∩X(74)X(759)

Trilinears    a^9-(b+c)*a^8-2*(b-c)^2*a^7+(b+c)*(3*b^2-5*b*c+3*c^2)*a^6-b*c*(6*b^2-11*b*c+6*c^2)*a^5-3*(b^3+c^3)*(b-c)^2*a^4+2*(b^4+c^4+2*b*c*(b^2+c^2))*(b-c)^2*a^3+(b^2-c^2)*(b-c)*(b^4+c^4-b*c*(b^2-4*b*c+c^2))*a^2-(b^2-c^2)^2*(b^4+c^4-b*c*(2*b-c)*(b-2*c))*a-(b^2-c^2)^3*(b-c)*b*c : :
X(14127) = (3*r^2 + 4*R^2 - s^2 + 4*R*r)*X(3) - R*(R - 2*r)*X(4)

As a point on the Euler line, X(14127) has Shinagawa coefficients (4r^2 - 2F, 8rR - 4r^2 - E + 2F).

See Tran Quang Hung and César Lozada, Advanced Plane Geometry 3912.

X(14127) lies on these lines: {2,3}, {74,759}, {104,900}, {477,12030}, {915,1295}, {953,2716}

X(14127) = reflection of X(i) in X(j) for these (i,j): (4,867), (13589,3)
X(14127) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (140, 431, 1656), (468, 6983, 7501), (1592, 8359, 13154), (4243, 6617, 6898), (5192, 10299, 12108), (6097, 6906, 6930), (6918, 10257, 13745), (6946, 10021, 5020), (7428, 11308, 11484), (7442, 7456, 7429), (7485, 7866, 6143), (7557, 11328, 6855), (10989, 14014, 3538)

X(14128) =  COMPLEMENT OF X(13630)

Barycentrics    a^2*((b^2+c^2)*a^2-(b^2-c^2)^2)*(a^4-2*(b^2+c^2)*a^2+5*b^2*c^2+c^4+b^4) : :
X(14128) = 3*X(2)+X(5876) = 9*X(3)-X(12279) = 7*X(3)+X(12290) = 5*X(4)+3*X(13340) = 7*X(5)-3*X(51) = 5*X(5)-X(52) = 3*X(5)-X(143) = 3*X(5)+X(5562) = 15*X(51)-7*X(52) = 9*X(51)-7*X(143) = 9*X(51)+7*X(5562) = 5*X(10627)-3*X(13340) = 7*X(12279)+9*X(12290)

See Tran Quang Hung and César Lozada, Advanced Plane Geometry 3925.

X(14128) lies on these lines: {2, 5876}, {3, 6030}, {4, 10627}, {5, 51}, {30, 5447}, {110, 10610}, {140, 5663}, {156, 7514}, {185, 632}, {381, 6101}, {382, 7999}, {389, 547}, {511, 3850}, {546, 1216}, {548, 3819}, {549, 12162}, {568, 5056}, {631, 13491}, {1493, 13434}, {1656, 6102}, {1657, 7998}, {2063, 9818}, {2979, 3843}, {3060, 5072}, {3090, 5946}, {3091, 10263}, {3526, 12111}, {3530, 6000}, {3534, 11439}, {3545, 6243}, {3567, 5079}, {3627, 3917}, {3628, 10219}, {3845, 10625}, {3851, 11412}, {4550, 11250}, {5054, 6241}, {5055, 5889}, {5066, 5446}, {5070, 5890}, {5609, 7550}, {5650, 10575}, {5943, 12046}, {7564, 10516}, {7723, 11561}, {8703, 11381}, {10024, 12358}, {10110, 12811}, {12103, 13474}, {12134, 13470}

X(14128) = reflection of X(i) in X(j) for these (i,j): (546, 11017), (548, 11592), (10095, 5), (10110, 12811), (12006, 3628)
X(14128) = complement of X(13630)
X(14128) = X(5)-of-X(5)-Brocard triangle
X(14128) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 5876, 13630), (5, 52, 13364), (5, 5562, 143), (5, 5891, 11591), (143, 11591, 5562), (381, 11444, 6101), (548, 3819, 11592), (1656, 6102, 13363), (1656, 11459, 6102), (5907, 10170, 140), (5943, 12812, 12046)

X(14129) =  POLAR CONJUGATE OF X(252)

Barycentrics    (S^2+SB*SC)*(SA^2-3*S^2)*SB*SC : :

See Tran Quang Hung and César Lozada, Advanced Plane Geometry 3939.

X(14129) lies on these lines: {2, 10979}, {4, 569}, {53, 311}, {94, 2052}, {275, 11538}, {472, 8836}, {473, 8838}, {1568, 6750}, {1625, 1993}, {3078, 10003}

X(14129) = polar conjugate of X(252)
X(14129) = {X(53), X(467)}-harmonic conjugate of X(324)

X(14130) =  EULER LINE INTERCEPT OF X(36)X(9628)

Barycentrics    a^2 (a^8 - 2 a^6 (b^2+c^2) + 5 a^4 b^2 c^2 + a^2 (2 b^6-b^4 c^2-b^2 c^4+2 c^6) - (b^2-c^2)^2 (b^4+4 b^2 c^2+c^4)) : :

Let HaHbHc be as at X(14709). Then X(14130) is the harmonic center of the circumcircles of ABC and HaHbHc. (Randy Hutson, December 2, 2017)

See Tran Quang Hung and Angel Montesdeoca, Advanced Plane Geometry 3941.

X(14130) lies on these lines: {2, 3}, {36, 9628}, {49, 399}, {54, 5663}, {74, 13434}, {185, 567}, {265, 13403}, {511, 12307}, {568, 7689}, {569, 3357}, {575, 5621}, {1092, 4550}, {2777, 3521}, {2935, 9730}, {3167, 9938}, {3567, 11454}, {3581, 5446}, {5012, 13491}, {5609, 9705}, {6000, 10274}, {6102, 11440}, {7691, 13391}, {8254, 11805}, {9655, 9672}, {9659, 9668}, {9703, 11441}, {10540, 13367}, {10564, 11793}, {10628, 11560}, {10982, 13321}, {11439, 11464}, {12006, 12041}, {12038, 12302}

X(14130) = reflection of X(13564) in X(3)
X(14130) = midpoint of X(14709) and X(14710)

X(14131) =  MIDPOINT OF X(3) AND X(5482)

Barycentrics    a (a^4 (b^2-6 b c+c^2) + a^3 (b^3+b^2 c+b c^2+c^3) - a^2 (b^4-7 b^3 c-7 b c^3+c^4) - a(b^5+b^4 c+b c^4+c^5) - b c (b^2-c^2)^2) : :
X(14131) = (r^2 + 2rR + s^2)*X(1) - (r^2 - 14rR + s^2)*X(3)

See Tran Quang Hung and Angel Montesdeoca, Advanced Plane Geometry 3950.

X(14131) lies on these lines: {1,3}, {511,3530}, {3524,5752}

X(14131) = midpoint of X(3) and X(5482)

X(14132) =  (name pending)

Barycentrics    a (a^3 (b+c)-a (b-c)^2 (b+c)-(b^2-c^2)^2+a^2 (b^2+c^2)) (a^5+a^4 (b+c)+(b-c)^2 (b+c)^3-a^3 (2 b^2+b c+2 c^2)-2 a^2 (b^3+b^2 c+b c^2+c^3)+a (b^4+b^3 c+2 b^2 c^2+b c^3+c^4)) : :

See Tran Quang Hung and Angel Montesdeoca, Advanced Plane Geometry 3960.

X(14132) lies on these lines: X(14132) lies on these lines: {1,3}, {5164,5948}

X(14132) = orthocenter of cevian triangle of X(1029)

X(14133) =  X(3)X(695)∩X(4)X(83)

Barycentrics    a^2 (-a^6 (b^2+c^2)+3 a^2 b^2 c^2 (b^2+c^2)+a^4 (b^4+c^4)+b^2 c^2 (b^4+c^4)) : :
X(14133) = 2*X(3)+X(31989)

See Tran Quang Hung and Angel Montesdeoca, Advanced Plane Geometry 3966.

X(14133) lies on these lines: {3,695}, {4,83}, {30,31869}, {110,33021}, {184,7791}, {511,23642}, {3203,7830}, {3491,20775}, {3796,11325}, {5012,6655}, {5092,6310}, {5907,13334}, {6656,36213}, {9306,16043}, {10323,35424}, {13336,37348}

X(14133) = midpoint of X(14134) and X(31989)
X(14133) = reflection of X(14134) in X(3)

X(14134) =  X(3)X(695)∩X(20)X(1352)

Barycentrics    a^2 (a^6 (b^2+c^2)+a^4 (b^4+c^4)-b^2 c^2 (b^4+4 b^2 c^2+c^4)-a^2 (2 b^6+b^4 c^2+b^2 c^4+2 c^6)) : :
X(14134) = 4*X(3)-X(31989)

See Tran Quang Hung and Angel Montesdeoca, Advanced Plane Geometry 3966.

X(14134) lies on these lines: {3,695}, {20,1352}, {140,31869}, {182,38834}, {511,8152}, {1799,37889}, {3492,28724}, {3819,11326}, {6310,7467}, {14135,14810}, {33269,34417}

X(14134) = reflection of X(i) in X(j) for these (i,j): (14133, 3), (31869, 140), (31989, 14133)


X(14135) =  X(1)X(7153)∩X(3)X(3229)

Barycentrics    a^2 (a^4 (b^4-4 b^2 c^2+c^4)+b^2 c^2 (b^4-4 b^2 c^2+c^4)-a^2 (b^6-4 b^4 c^2-4 b^2 c^4+c^6)) : :
X(14135) = X(20)+3*X(35687)

See Tran Quang Hung and Angel Montesdeoca, Advanced Plane Geometry 3966v.

X(14135) lies on these lines: {1,7153}, {3,3229}, {20,185}, {39,6234}, {51,6658}, {99,3491}, {182,3224}, {384,34236}, {575,9490}, {3051,31989}, {3819,32965}, {3917,33260}, {5167,7782}, {5943,14035}, {6179,32442}, {6688,32971}, {7748,35060}, {8841,9737}, {12006,33962}, {12038,30209}, {14134,14810}, {15082,33258}, {15488,34020}, {19691,33873}, {21849,33193}, {27374,33250}

X(14135) = reflection of X(6310) in X(3)
X(14135) = X(4)-of-tangential-triangle-of-hyperbola {A,B,C,X(2),X(6)}
X(14135) = Euler-Gergonne-Soddy-circle-inverse of X(33677)

X(14136) =  MIDPOINT OF X(13) AND X(61)

Barycentrics    sqrt(3)*(2*(SA-2*SW)*S^2+(SB+ SC)*SA*SW)-S*(2*S^2-3*SW*SA+3* SW^2) : :

See Le Viet An and César Lozada, Hyacinthos 26514.

+

X(14136) lies on these lines: {4, 13}, {6, 5617}, {17, 299}, {30, 10613}, {62, 618}, {114, 9300}, {115, 6783}, {396, 511}, {397, 8259}, {530, 11299}, {533, 5459}, {5613, 6034}, {6772, 13103}, {11129, 11289}

X(14136) = midpoint of X(13) and X(61)
X(14136) = reflection of X(i) in X(j) for these (i,j): (618, 6694), (635, 6669)
X(14136) = {X(6), X(6115)}-harmonic conjugate of X(6782)

X(14137) =  MIDPOINT OF X(14) AND X(62)

Barycentrics    sqrt(3)*(2*(SA-2*SW)*S^2+(SB+ SC)*SA*SW)+S*(2*S^2-3*SW*SA+3* SW^2) : :

See Le Viet An and César Lozada, Hyacinthos 26514.

X(14137) lies on these lines: {4, 14}, {6, 5613}, {18, 298}, {30, 10614}, {61, 619}, {114, 9300}, {115, 6782}, {395, 511}, {398, 8260}, {531, 11300}, {532, 5460}, {5617, 6034}, {6775, 13102}, {11128, 11290}

X(14137) = midpoint of X(14) and X(62)
X(14137) = reflection of X(i) in X(j) for these (i,j): (619, 6695), (636, 6670)
X(14137) = {X(6), X(6114)}-harmonic conjugate of X(6783)

X(14138) =  MIDPOINT OF X(15) AND X(17)

Barycentrics    (2*S^2+(SW+2*sqrt(3)*S)*(SB+ SC))*(SA+sqrt(3)*S) : :

See Le Viet An and César Lozada, Hyacinthos 26514.

X(14138) lies on these lines: {4, 15}, {30, 10611}, {61, 302}, {114, 6109}, {193, 627}, {396, 13350}, {511, 8259}, {532, 5463}, {623, 6673}, {1570,14139}

X(14138) = midpoint of X(15) and X(17)
X(14138) = reflection of X(i) in X(j) for these (i,j): (623, 6673), (629, 6671)

X(14139) =  MIDPOINT OF X(16) AND X(18)

Barycentrics    (2*S^2+(SW-2*sqrt(3)*S)*(SB+ SC))*(SA-sqrt(3)*S) : :

See Le Viet An and César Lozada, Hyacinthos 26514.

X(14139) lies on these lines: {4, 16}, {30, 10612}, {62, 303}, {114, 6108}, {193, 628}, {395, 13349}, {511, 8260}, {533, 5464}, {624, 6674}, {1570,14138}

X(14139) = midpoint of X(16) and X(18)
X(14139) = reflection of X(i) in X(j) for these (i,j): (624, 6674), (630, 6672)

X(14140) =  X(5)X(51)∩X(30)X(930)

Trilinears    cos(B-C)*(4*(2*cos(A)+cos(3*A) )*cos(B-C)+2*cos(2*A)*cos(2*( B-C))-cos(4*A)+1/2) : :
X(14140) = 3*X(549) - 2*X(6150)

See Le Viet An and César Lozada, Hyacinthos 26516.

X(14140) lies on these lines: {5, 51}, {30, 930}, {140, 1157}, {549, 6150}, {632, 10615}, {5432, 14102}, {7604, 11016}

X(14140) = reflection of X(1157) in X(140)
X(14140) = ninepoint-circle-inverse of X(13565)

X(14141) =  X(5)X(51)∩X(30)X(13512)

Trilinears    cos(B-C)*(8*(2*cos(A)+cos(3*A) )*cos(B-C)+2*cos(2*A)*cos(2*( B-C))-3*cos(4*A)-1/2) : :
X(14141) = 3*X(549) - 2*X(1157)

See Le Viet An and César Lozada, Hyacinthos 26516.

X(14141) lies on these lines: {5, 51}, {30, 13512}, {549, 1157} , {10615, 11539}

X(14142) =  REFLECTION OF X(4) IN X(5501)

Trilinears    (5*cos(2*A)+5*cos(4*A)+3/2)* cos(B-C)-2*(3*cos(A)+cos(3*A)) *cos(2*(B-C))+cos(2*A)*cos(3*( B-C))-cos(3*A)-cos(5*A)-7*cos( A) : :
X(14142) = (5*R^2-2*SW)^2*X(3) - (9*R^4-8* SW*R^2-2*S^2+2*SW^2)*X(5)

See Le Viet An and César Lozada, Hyacinthos 26516.

X(14142) lies on this line: {2,3}

X(14142) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (964, 7475, 13628), (1370, 7475, 11331), (3090, 7460, 7487), (3533, 6990, 7435), (4244, 7715, 3850), (7442, 11345, 6955), (7473, 10223, 462)

X(14143) =  X(3)X(2888)∩X(5)X(252)

Trilinears    (cos(2*A)-cos(4*A)+7/2)*cos(B- C)+(2*cos(A)-2*cos(3*A))*cos( 2*(B-C))+(cos(2*A)+1)*cos(3*( B-C))-cos(5*A)+cos(A)-cos(3*A) : :

See Le Viet An and César Lozada, Hyacinthos 26522.

X(14143) lies on these lines: {3, 2888}, {4, 14051}, {5, 252}, {1209, 6150}

X(14143) = {X(3), X(2888)}-harmonic conjugate of X(14072)

X(14144) =  DUAL OF X(5613)

Barycentrics    Sqrt(3) (S^2 SA + SB SC SW) + S (3 SA^2 + 6 SB SC - 5 S^2) : :      (Peter Moses, August 27, 2017)
X(14144) = 4 X(619) - X(11122), X(622) - 4 X(11132)    (Peter Moses)

Let BA'C be the equilateral triangle with side BC and A' on the side of BC that does not include A. Let La be the line through A' parallel to B'C', and define Lb and Lc cyclically. Let A'' = Lb∩Lc, and define B''and C'' cyclically. The Euler lines of BA"C, CB''A, AC''B concur in X(14144). The acute angle between each pair of these Euler lines is π/6. See Dao Thanh Oai, ADGEOM 4032, and Francisco Javier García Capitán, ADGEOM 4033. See also X(5617) and X(14145).

X(14144) lies on these lines: {2,5469}, {3,298}, {14,629}, {17,619}, {99,622}, {532,5464}, {628,10653}, {1352,14145}, {5474,5978}, {8594,11154}, {8716,9763}, {9736,11129}

X(14144) = midpoint of X(617) and X(627)
X(14144) = reflection of X(i) in X(j) for these {i,j}: {14, 629}, {17, 619}, {11122, 17}

X(14145) =  DUAL OF X(5617)

Barycentrics    Sqrt(3) (S^2 SA + SB SC SW) - S (3 SA^2 + 6 SB SC - 5 S^2) : :      (Peter Moses, August 27, 2017)
X(14145) = 4 X(618) - X(11121), X(621) - 4 X(11133)    (Peter Moses)

Let BA'C be the equilateral triangle with side BC and A' on the side of BC that includes A. Let La be the line through A' parallel to B'C', and define Lb and Lc cyclically. Let A'' = Lb∩Lc, and define B''and C'' cyclically. The Euler lines of BA"C, CB''A, AC''B concur in X(14145). The acute angle between each pair of these Euler lines is π/6. See Dao Thanh Oai, ADGEOM 4032, and Francisco Javier García Capitán, ADGEOM 4033. See also X(5613) and X(14144).

X(14145) lies on these lines: {2,5470}, {3,299}, {13,630}, {18,618}, {99,621}, {533,5463}, {627,10654}, {1352,14144}, {5473,5979}, {8595,11153}, {8716,9761}, {9735,11128}

X(14145) = midpoint of X(616) and X(628)
X(14145) = reflection of X(i) in X(j) for these {i,j}: {13, 630}, {18, 618}, {11121, 18}

X(14146) = ISOGONAL CONJUGATE OF X(5637)

Trilinears    csc((B-C)/3) : :

The perspectors of ABC and the 1st Morley adjunct triangle, the 2nd Morley adjunct triangle, and the 3rd Morley adjunct triangle are collinear on the trilinear polar of X(14146).

X(14146) lies on these lines: {15,3334}, {61,3272}, {62,3335}, {3412,3609}

X(14146) = isogonal conjugate of X(5637)
X(14146) = trilinear pole of line X(16)X(358)

X(14147) =  (name pending)

Barycentrics    a (a-b) (a-c) (a^10 - a^9 (b+c) + a^8 (-3 b^2+5 b c-3 c^2) + a^7 (4 b^3-2 b^2 c-2 b c^2+4 c^3) + a^6 (2 b^4-5 b^3 c+5 b^2 c^2-5 b c^3+2 c^4) - a^5 (6 b^5-6 b^4 c+b^3 c^2+b^2 c^3-6 b c^4+6 c^5) + a^4 (2 b^6-3 b^5 c-4 b^4 c^2+11 b^3 c^3-4 b^2 c^4-3 b c^5+2 c^6) + a^3 (b-c)^2 (4 b^5+6 b^4 c+7 b^3 c^2+7 b^2 c^3+6 b c^4+4 c^5) - a^2 (b^2-c^2)^2 (3 b^4-b^3 c+b^2 c^2-b c^3+3 c^4) - a (b-c)^4 (b+c)^5 + (b-c)^4 (b+c)^6) : :

See Tran Quang Hung and Angel Montesdeoca, ADGEOM 3990

X(14147) lies on this line: {37,11075}

X(14148) = REFLECTION OF X(620) IN X(6390)

Barycentrics    2 a^4-4 a^2 b^2+b^4-4 a^2 c^2+4 b^2 c^2+c^4 : :
X(14148) = 2 X(230) - 3 X(620) = X(385) - 3 X(2482) = X(230) - 3 X(6390) = 3 X(99) - X(6781) = 3 X(99) + X(7779) = X(115) - 3 X(7799) = X(7779) - 3 X(7813) = X(6781) + 3 X(7813) = 3 X(115) - 5 X(7925) = 9 X(7799) - 5 X(7925) = 7 X(325) - 3 X(8352) = 5 X(5461) - 6 X(10150) = 3 X(9167) - X(11054) = X(5477) - 3 X(12215)

X(14148) is described in connection with a quartic curve and the asymptotes of the Kiepert hyperbola; see Bernard Gibert, Q140: a quartic through the X(5)-anticevian points.

X(14148) lies on these lines: {20,7916}, {39,6704}, {99,754}, {115,7799}, {194,5355}, {230,538}, {325,543}, {385,2482}, {550,7882}, {626,2549}, {1569,9865}, {1975,5475}, {3266,10418}, {3552,7890}, {3630,8703}, {3734,7736}, {3933,7830}, {4045,7801}, {5304,7798}, {5461,10150}, {5477,12215}, {6337,7751}, {7738,7869}, {7747,7906}, {7755,7891}, {7756,7796}, {7757,7820}, {7761,8716}, {7765,7836}, {7782,7826}, {7783,7794}, {7789,7829}, {7806,11055}, {7816,7838}, {7915,9607}, {9167,11054}

X(14148) = midpoint of X(i) and X(j) for these {i,j}: {99, 7813}, {1569, 9865}, {6781, 7779}
X(14148) = reflection of X(620) in X(6390)
X(14148) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (99, 7779, 6781), (194, 7835, 5355), (194, 7863, 6680), (2549, 3926, 7908), (2549, 7908, 626), (3926, 7781, 626), (5355, 7835, 6680), (5355, 7863, 7835), (6781, 7813, 7779), (7781, 7908, 2549)

X(14149) =  (name pending)

Barycentrics    (-a^2+b^2-c^2) (a^2+b^2-c^2) (a^4-2 a^2 b^2+b^4-2 a^2 c^2+c^4) (-a^2 b^2+b^4-a^2 c^2-2 b^2 c^2+c^4) (-a^6 b^2+3 a^4 b^4-3 a^2 b^6+b^8-2 a^6 c^2+a^4 b^2 c^2+4 a^2 b^4 c^2-3 b^6 c^2+4 a^4 c^4+a^2 b^2 c^4+3 b^4 c^4-2 a^2 c^6-b^2 c^6) (2 a^6 b^2-4 a^4 b^4+2 a^2 b^6+a^6 c^2-a^4 b^2 c^2-a^2 b^4 c^2+b^6 c^2-3 a^4 c^4-4 a^2 b^2 c^4-3 b^4 c^4+3 a^2 c^6+3 b^2 c^6-c^8) : :

See Tran Quang Hung and Angel Montesdeoca, Hyacinthos 26537.

X(14149) lies on these lines: {4,216}, {52,11547}, {324,6750}

X(14150) =  (name pending)

Barycentrics    a*(2*a^3+(b+c)*a^2-2*(b^2+13* b*c+c^2)*a-(b+c)*(b^2+10*b*c+ c^2)) : :
X(14150) = X(1) + 3*X(3646) = 5*X(1) + 3*X(4866) = X(3296) - 5*X(3616) = 5*X(3646) - X(4866)

See Antreas Hatzipolakis and César Lozada, Hyacinthos 26540.

X(14150) lies on these lines:
{1, 210}, {551, 5302}, {1001, 13624}, {1125, 3579} , {3296, 3616}, {3634, 3813}, {3647, 11281}, {5550, 11024}, {5919, 11524}

X(14151) =  X(1)X(651)∩X(7)X(528)

Barycentrics    a*(a^3-4*(b+c)*a^2+(5*b^2+3*b* c+5*c^2)*a-(b+c)*(2*b^2-b*c+2* c^2))*(a-b+c)*(a+b-c) : :
X(14151) = 4*X(1) - X(1156) = X(100) + 2*X(3243) = 4*X(142) - X(12531) = X(390) - 4*X(12735) = 4*X(5083) - X(7672)

See Le Viet An and César Lozada, Hyacinthos 26542.

X(14151) lies on these lines:
{1, 651}, {7, 528}, {11, 3475}, {57, 100}, {59, 518}, {104, 2346}, {142, 12531}, {145, 10427}, {214, 1445}, {226, 10707}, {390, 6938}, {484, 10087}, {497, 12831}, {952, 1056}, {1145, 8732}, {1387, 8232}, {1388, 5220}, {1420, 6594}, {1617, 4430}, {2800, 7675}, {2802, 4321}, {3254, 10106}, {3315, 4551}, {3340, 5528}, {3600, 10609}, {3748, 10391}, {3889, 13279}, {4318, 4864}, {4326, 13253}, {4532, 5223}, {4861, 5784}, {5045, 12738}, {5119, 7676}, {5261, 12019}, {5289, 6172}, {5330, 5698}, {5425, 5542}, {5435, 6174}, {5572, 12740}, {5660, 11019}, {5722, 10711}, {5728, 6265}, {5732, 7673}, {5851, 8236}, {6049, 6068}, {6264, 9846}, {6326, 11025}, {7679, 10956}, {7982, 8544}, {8098, 8389}, {8104, 11234}, {8388, 12748}, {11219, 13405}, {12750, 13407}

X(14151) = anticomplement of X(38211)
X(14151) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (7, 1317, 12730), (1317, 3476, 10031)

X(14152) =  X(3)X(49) ∩X(30)X(1105)

Trilinears    cos(A)*(4*cos(A)*cos(B-C)-cos( 2*(B-C))-cos(2*A)+cos(4*A)-3) : :
X(14152) = (24*R^4-10*R^2*SW+S^2+SW^2)*X( 3) - 2*(7*R^2-2*SW)*(4*R^2-SW)* X(49)

See Antreas Hatzipolakis and César Lozada, Hyacinthos 26544.

X(14152) lies on these lines:
{3, 49}, {4, 2055}, {30, 1105}, {417, 6760}, {418, 1614}, {426, 14059}, {542, 10600}, {577, 6759}, {933, 8439}, {1629, 13322}, {3284, 10110}, {6638, 10539}, {6641, 7592}, {9225, 9243}

X(14152) = T-isogonal conjugate of X(3), where T = cevian triangle of X(3)

X(14153) =  X(6)X(25) ∩X(54)X(695)

Trilinears    a*(a^4-2*(b^2+c^2)*a^2-b^2*c^ 2) : :
X(14153) = SW*(3*S^2+SW^2)*X(6) - 2*S^2*(6* R^2-SW)*X(25)

See Antreas Hatzipolakis and César Lozada, Hyacinthos 26544.

X(14153) lies on these lines:
{2, 2056}, {6, 25}, {22, 13330}, {54, 695}, {111, 12834}, {182, 1613}, {323, 8041}, {511, 10329}, {524, 1799}, {575, 1196}, {1180, 11422}, {1207, 3203}, {1501, 11003}, {1627, 1691}, {1993, 3094}, {1994, 5111}, {2001, 13331}, {3117, 3398}, {3787, 5092}, {3796, 5017}, {3917, 5116}, {4074, 12215}, {5034, 9306}, {5104, 6636}, {5133, 11646}, {13410, 13595}

X(14153) = T-isogonal conjugate of X(6), where T = cevian triangle of X(6)
X(14153) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 2056, 9225), (6, 184, 1915), (6, 9604, 1971), (1194, 13366, 6), (2056, 5038, 2), (3051, 5012, 1691)

X(14154) =  (name pending)

Barycentrics    4*a^6-8*(b+c)*a^5+11*b*c*a^4+( b+c)*(8*b^2-11*b*c+8*c^2)*a^3- (4*b^2+9*b*c+4*c^2)*(b-c)^2*a^ 2-(b^2-c^2)*(b-c)*b*c*a+2*b*c* (b-c)^4 : :
X(14154) = (4s^4+8(5R^2-SW)s^2+5RSs-2(16R^2-r^2-2SW)SW)*X(7) + 4(S+2sR)(S-sR)*X(1155)

See Antreas Hatzipolakis and César Lozada, Hyacinthos 26546.

X(14154) lies on this line: {7,1155}

X(14154) = T-isogonal conjugate of X(7), where T = circumcevian triangle of X(7)

X(14155) =  (name pending)

Trilinears    (p^3*(16*p^3-16*p^2*q-5*p+20* q)-(9*q^2+7)*p^2+2*(q^2+2)*q* p-q^2)/p^2 : : , where p=sin(A/2), q=cos((B-C)/2) : :

See Antreas Hatzipolakis and César Lozada, Hyacinthos 26546.

X(14155) lies on this line: {8,1319}

X(14155) = T-isogonal conjugate of X(8), where T = circumcevian triangle of X(8)

X(14156) = X(125)X(539) ∩ X(140)X(389)

Barycentrics    (2*a^8-4*(b^2+c^2)*a^6+(b^4+4*b^2*c^2+c^4)*a^4+2*(b^4-c^4)*(b^2-c^2)*a^2-(b^2-c^2)^4)*(-a^2+b^2+c^2) : :
X(14156) = X(6699)+2*X(11064)

X(14156) is described in relation with Bernard Gibert curve Q141. (César Lozada, Aug. 26, 2017)

X(14156) lies on these lines: {2,13352}, {3,1568}, {5,12897}, {30,5972}, {113,2071}, {125,539}, {140,389}, {399,13399}, {403,10564}, {597,5097}, {974,6699}, {1092,5449}, {1147,1899}, {5642,10540}, {11449,11750}

X(14156) = midpoint of X(i) and X(j) for these {i,j}: {3, 1568}, {113, 2071}, {399, 13399}, {403, 10564}, {10257, 11064}
X(14156) = reflection of X(i) in X(j) for these (i,j): (403, 12900), (6699, 10257)
X(14156) = {X(1092), X(6640)}-harmonic conjugate of X(5449)

X(14157) = X(4)X(54) ∩ X(30)X(110)

Trilinears    2a sec A cos(B - C) + (b sec A - a sec B) cos(C - A) + (c sec A - a sec C) cos(A - B) - (b sec C + c sec B) cos(A - B) cos(C - A) sec(B - C) : :
Barycentrics    (a^8-3*(b^2+c^2)*a^6+(3*b^4-b^2*c^2+3*c^4)*a^4-(b^4-c^4)*(b^2-c^2)*a^2+3*(b^2-c^2)^2*b^2*c^2)*a^2 : :
X(14157) = 2*X(23)+X(14094) = X(74)-4*X(1495) = X(74)+2*X(12112) = 2*X(1495)+X(12112) = 2*X(1533)+X(12383) = X(10752)+2*X(12367)

X(14157) is described in relation with Bernard Gibert curve Q141. (César Lozada, Aug. 26, 2017)

Let P1 and P2 be the two points on the circumcircle whose Steiner lines are tangent to the circumcircle. Let Q1 and Q2 be the respective points of tangency. Then X(14157) = P1Q1∩P2Q2. (Randy Hutson, November 2, 2017)

X(14157) lies on these lines: {3,6030}, {4,54}, {20,10539}, {22,11459}, {23,13754}, {24,1192}, {25,5890}, {26,12111}, {30,110}, {34,9638}, {49,3627}, {64,11468}, {74,186}, {112,1971}, {113,3153}, {154,378}, {156,382}, {182,3545}, {185,3518}, {232,13509}, {265,11563}, {376,9306}, {381,5012}, {394,12082}, {399,1154}, {403,1503}, {546,13434}, {547,13339}, {567,3845}, {569,3832}, {1092,3529}, {1147,3146}, {1157,1510}, {1173,1199}, {1176,3818}, {1177,11564}, {1181,3567}, {1533,12383}, {1593,9707}, {1596,12022}, {1598,7592}, {1625,10313}, {1658,11440}, {1870,10535}, {2070,5663}, {2328,7430}, {2360,7421}, {2777,13619}, {2883,6240}, {2914,11807}, {2937,5876}, {2979,12083}, {3043,13202}, {3047,12295}, {3060,7530}, {3090,10984}, {3147,12324}, {3426,11410}, {3431,13603}, {3515,12315}, {3517,12174}, {3520,10282}, {3524,5651}, {3533,13347}, {3542,11457}, {3543,9544}, {3830,13482}, {3839,11003}, {3850,13353}, {3853,9706}, {4550,7712}, {5056,13336}, {5076,9704}, {5198,11423}, {5318,11137}, {5321,11134}, {5494,6001}, {5562,12088}, {5609,13391}, {5654,7391}, {5889,7517}, {5891,6636}, {5907,7512}, {5944,14130}, {5946,7545}, {6247,10018}, {6288,10203}, {6747,6750}, {6800,9818}, {7387,11412}, {7488,12162}, {7506,10574}, {7526,11439}, {7577,11550}, {7999,10323}, {8717,10304}, {8744,8779}, {8972,9687}, {9652,12953}, {9667,12943}, {9730,13595}, {9970,11649}, {10096,10264}, {10117,12281}, {10602,12283}, {10625,12087}, {10632,10676}, {10633,10675}, {10752,12367}, {10880,12970}, {10881,12964}, {11430,13596}, {11449,12084}, {11591,13564}, {12038,12086}, {12316,13421}, {12834,13364}, {13198,13851}, {13367,13474}

X(14157) = midpoint of X(i) and X(j) for these {i,j}: {186, 12112}, {399, 5899}
X(14157) = reflection of X(i) in X(j) for these (i,j): (74, 186), (110, 10540), (186, 1495), (265, 11563), (3153, 113), (10264, 10096), (13445, 3)
X(14157) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4, 1614, 54), (4, 9833, 12289), (4, 12254, 13403), (24, 1498, 6241), (25, 11456, 5890), (154, 378, 11464), (1181, 10594, 3567), (1199, 10110, 1173), (1495, 12112, 74), (1598, 7592, 9781), (1971, 3331, 112), (11455, 11464, 378)

X(14158) =  (name pending)

Barycentrics    a (a^3+a^2 (b+c)-(b-c)^2 (b+c)-a (b^2+b c+c^2))/((b+c) (-a^2+b^2+b c+c^2)) : :

See Tran Quang Hung and Angel Montesdeoca, ADGEOM 3997

X(14158) = barycentric product X(86)*X(11076)
X(14158) = barycentric quotient X(i)/X(j) for these (i, j): (484, 3969), (1333, 7343)
X(14158) = trilinear product X(81)*X(11076)
X(14158) = trilinear quotient X(i)/X(j) for these (i, j): (58, 7343), (484, 3678)
X(14158) = X(10)-isoconjugate-of-X(7343)
X(14158) = X(i)-reciprocal conjugate of-X(j) for these (i,j): (484, 3969), (1333, 7343)

X(14159) =  (name pending)

Barycentrics    2*a^8-12*(b^2+c^2)*a^6+(29*b^4+20*b^2*c^2+29*c^4)*a^4-(b^2+c^2)*(27*b^4-53*b^2*c^2+27*c^4)*a^2+(8*b^4-17*b^2*c^2+8*c^4)*(b^2-c^2)^2 : :
X(14159) = (39*S^2 - SW^2)*X(3) + (33*S^2 + SW^2)*X(4)

See Tran Quang Hung and César Lozada, ADGEOM 4001

X(14159) lies on this line: {2,3}

X(14159) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (24, 5066, 10011), (421, 1113, 6973), (547, 5066, 10011), (4196, 8364, 1559), (4246, 8356, 5079), (6873, 7388, 7461)

X(14160) = (name pending)

Barycentrics    2*a^8-3*(b^2+c^2)*a^6-6*(b^4+b^2*c^2+c^4)*a^4+(b^2+c^2)*(11*b^4-20*b^2*c^2+11*c^4)*a^2-4*(b^2-c^2)^4 : :
X(14160) = 7*X(3851) - X(5171)

See Tran Quang Hung and César Lozada, ADGEOM 4001

X(14160) lies on these lines:
{4, 7769}, {5, 7830}, {30, 7619}, {115, 5097}, {381, 511}, {575, 5475}, {3091, 7898}, {3830, 9734}, {3843, 9737}, {3851, 5171}, {7694, 11645}

X(14160) = midpoint of X(3830) and X(9734)

X(14161) =  (name pending)

Barycentrics    5*a^8-24*(b^2+c^2)*a^6+(47*b^4+32*b^2*c^2+47*c^4)*a^4-(b^2+c^2)*(39*b^4-77*b^2*c^2+39*c^4)*a^2+(11*b^4-23*b^2*c^2+11*c^4)*(b^2-c^2)^2 : :
X(14161) = (63*S^2-SW^2)*X(3) + (45*S^2+SW^2)*X(4)

See Tran Quang Hung and César Lozada, ADGEOM 4003

X(14161) lies on this line: {2,3}

X(14161) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (471, 1889, 5054), (632, 3540, 2566), (1344, 6934, 11290), (4245, 13737, 14038), (7405, 7522, 7453), (7444, 13861, 11818), (7488, 7892, 14119), (11323, 11346, 11585)

: :

X(14162) = (name pending)

Barycentrics    5*(b^2+c^2)*a^6-2*(9*b^4+7*b^2*c^2+9*c^4)*a^4+(b^2+c^2)*(19*b^4-36*b^2*c^2+19*c^4)*a^2-6*(b^2-c^2)^4 : :
X(14162) = 13*X(5079) - X(5171)

See Tran Quang Hung and César Lozada, ADGEOM 4003

X(14162) lies on these lines:
{5, 141}, {381, 9734}, {5072, 9737}, {5079, 5171}, {7603, 11171}

X(14163) =  MOSES-YFF IMAGE OF X(111)

Barycentrics    (a^6-a^4 b^2+a^2 b^4+b^6-a^4 c^2-a^2 b^2 c^2-3 b^4 c^2+a^2 c^4+3 b^2 c^4-c^6) (a^6-a^4 b^2+a^2 b^4-b^6-a^4 c^2-a^2 b^2 c^2+3 b^4 c^2+a^2 c^4-3 b^2 c^4+c^6) : :

If P = p : q : r (barycentrics) lies on the circumcircle, then the following point, introduced here as the Moses-Yff image of P, lies on the Yff hyperbola:

Y(P) = b^2 c^2 (a^4+a^2 b^2-2 b^4-2 a^2 c^2+4 b^2 c^2-2 c^4) (a^4-2 a^2 b^2-2 b^4+a^2 c^2+4 b^2 c^2-2 c^4) p-a^2 (2 a^4-a^2 b^2-b^4-a^2 c^2+2 b^2 c^2-c^4) (c^2 (-a^2 b^2+(a^2+b^2-c^2)^2) q+b^2 (-a^2 c^2+(a^2-b^2+c^2)^2) r) : : . For example, the Moses-Yff image of X(110) is X(2), and that of X(74) is X(4).     (Peter Moses, August 30, 2017)

Suppose that P is a point on the Yff hyperbola and, that P is not X(2) or X(4). Then the following points are also on the Yff hyperbola:

reflection of P in X(381)
reflection of P in the Euler line
reflection of P in the line through X(381) perpendicuar to the Euler line. Thus, the four points are the vertices of a rectangle inscribed in the Yff hyperbola.     (Peter Moses, August 30, 2017)

Suppose that P = p : q : r (barycentrics) is a point on the circumcircle. Then the four points given as follows by barycentric corrdinates are vertices of a rectangle inscribed in the Yff hyperbola:     (Peter Moses, August 30, 2017)

b^2 c^2 (a^4+a^2 b^2-2 b^4-2 a^2 c^2+4 b^2 c^2-2 c^4) (a^4-2 a^2 b^2-2 b^4+a^2 c^2+4 b^2 c^2-2 c^4) p-a^2 (2 a^4-a^2 b^2-b^4-a^2 c^2+2 b^2 c^2-c^4) (c^2 (-a^2 b^2+(a^2+b^2-c^2)^2) q+b^2 (-a^2 c^2+(a^2-b^2+c^2)^2) r) : :

b^2 c^2 (a^2-a b+b^2-c^2) (a^2+a b+b^2-c^2) (a^2-b^2-a c+c^2) (a^2-b^2+a c+c^2) p-a^2 c^2 (a^4-2 a^2 b^2+b^4+a^2 c^2+b^2 c^2-2 c^4) (2 a^4+2 a^2 b^2-b^4-4 a^2 c^2-b^2 c^2+2 c^4) q-a^2 b^2 (2 a^4-4 a^2 b^2+2 b^4+2 a^2 c^2-b^2 c^2-c^4) (a^4+a^2 b^2-2 b^4-2 a^2 c^2+b^2 c^2+c^4) r : :

a^2 (a^2 b^2 c^2 (a^4-2 a^2 b^2+b^4+a^2 c^2+b^2 c^2-2 c^4) (a^4+a^2 b^2-2 b^4-2 a^2 c^2+b^2 c^2+c^4) p+c^2 (a^2-b^2-c^2) (a^2-a b+b^2-c^2) (a^2+a b+b^2-c^2) (a^4 b^2-2 a^2 b^4+b^6+a^4 c^2+2 a^2 b^2 c^2-b^4 c^2-2 a^2 c^4-b^2 c^4+c^6) q+b^2 (a^2-b^2-c^2) (a^2-b^2-a c+c^2) (a^2-b^2+a c+c^2) (a^4 b^2-2 a^2 b^4+b^6+a^4 c^2+2 a^2 b^2 c^2-b^4 c^2-2 a^2 c^4-b^2 c^4+c^6) r) : :

b^2 c^2 (-a^2+b^2-c^2) (a^2+b^2-c^2) (a^2-a b+b^2-c^2) (a^2+a b+b^2-c^2) (-a^2+b^2-a c-c^2) (-a^2+b^2+a c-c^2) p+a^2 b^2 c^2 (-2 a^4+a^2 b^2+b^4+a^2 c^2-2 b^2 c^2+c^4) (a^6-a^4 b^2-a^2 b^4+b^6-2 a^4 c^2+2 a^2 b^2 c^2-2 b^4 c^2+a^2 c^4+b^2 c^4) q+a^2 b^2 c^2 (-2 a^4+a^2 b^2+b^4+a^2 c^2-2 b^2 c^2+c^4) (a^6-2 a^4 b^2+a^2 b^4-a^4 c^2+2 a^2 b^2 c^2+b^4 c^2-a^2 c^4-2 b^2 c^4+c^6) r : :

For P = X(111), two of those points are X(14163) and X(14164), and the other two are X(14214) and X(14215).

If you have The Geometer's Sketchpad, you can view Yff Hyperbola with Inscribed Rectangle and Yff Hyperbola with Details.

If you have Geogebra, you can view X(14163) on the Yff hyperbola, including vertices X(14164, X(14214), X(1425) and center X(381). You can also view Yff Hyperbola with Rectangle.

Suppose that P is a point. Then the point with barycentrics shown here lies on the Yff hyperbola (Peter Moses, August 30, 2017) :

((a^4-2 a^2 b^2+b^4+a^2 c^2-c^4) p-(a^2-b^2) (a^2-b^2+c^2) q-(b^2-c^2) c^2 r)(-(a^4+a^2 b^2-b^4-2 a^2 c^2+c^4) p-b^2 (b^2-c^2) q+(a^2-c^2) (a^2+b^2-c^2) r) : :

X(14163) lies on these lines: {2,2453}, {6792,8029}

X(14163) = reflection of X(14214) in X(381)
X(14163) = reflection of X(14164) in the Euler line
X(14163) = reflection of X(14215) in line X(381)X(523)

X(14164) =  MOSES-YFF IMAGE OF X(9184)

Barycentrics    (a^6+a^4 b^2-a^2 b^4+b^6-3 a^4 c^2-a^2 b^2 c^2-b^4 c^2+3 a^2 c^4+b^2 c^4-c^6) (a^6-3 a^4 b^2+3 a^2 b^4-b^6+a^4 c^2-a^2 b^2 c^2+b^4 c^2-a^2 c^4-b^2 c^4+c^6) : :

See X(14163).

X(14164) lies on these lines: {2,6}, {4,8151}

X(14164) = reflection of X(14215) in X(381)
X(14164) = reflection of X(14163) in the Euler line
X(14164) = reflection of X(14214) in line X(381)X(523)

X(14165) =  X(15)X(470)∩X(16)X(471)

Trilinears    csc^2 2A sin 3A : :
Barycentrics    4 - sec^2 A : :
Barycentrics    3 - tan2A : :
Barycentrics    (a^2+b^2-c^2)^2 (a^2-b^2-b c-c^2) (a^2-b^2+b c-c^2) (a^2-b^2+c^2)^2 : :

See Bernard Gibert, K186.

X(14165) lies on the cubic K490 and these lines: {2,216}, {4,1495}, {15,470}, {16,471}, {25,9993}, {107,468}, {136,421}, {186,3258}, {249,297}, {275,1971}, {323,340}, {389,3462}, {403,1300}, {427,1629}, {436,6747}, {450,5972}, {451,1896}, {458,7790}, {648,3580}, {1093,7505}, {1514,10152}, {1585,6561}, {1586,6560}, {1594,8884}, {1748,1786}, {2207,3096}, {2373,6330}, {2963,8794}, {4232,10002}, {6619,11206}, {8779,11427}.

X(14165) = cevapoint of X(i) and X(j) for these (i,j): {403, 1990}
X(14165) = crosssum of X(i) and X(j) for these (i,j): {216, 3284}, {1636, 3269}
X(14165) = X(i)-cross conjugate of X(j) for these (i,j): {50, 5962}, {186, 340}, {11062, 186}
X(14165) = trilinear pole of line {526, 1986}
X(14165) = pole wrt polar circle of trilinear polar of X(265) (line X(216)X(647))
X(14165) = polar conjugate of X(265)
X(14165) = isoconjugate of X(j) and X(j) for these (i,j): {48, 265}, {255, 1989}, {326, 11060}, {328, 9247}, {476, 822}, {577, 2166}, {1820, 5961}, {2315, 12028}, {4100, 6344}
X(14165) = barycentric product X(i)X(j) for these {i,j}: {4, 340}, {107, 3268}, {186, 264}, {276, 11062}, {317, 5962}, {323, 2052}, {393, 7799}, {470, 471}, {526, 6528}, {1154, 8795}, {1273, 8884}, {5081, 7282}
X(14165) = barycentric quotient X(i)/X(j) for these {i,j}: {4, 265}, {24, 5961}, {50, 577}, {107, 476}, {158, 2166}, {186, 3}, {264, 328}, {323, 394}, {340, 69}, {393, 1989}, {526, 520}, {562, 3519}, {1093, 6344}, {1154, 5562}, {1300, 12028}, {1870, 7100}, {1986, 13754}, {2052, 94}, {2088, 3269}, {2207, 11060}, {2624, 822}, {3258, 1650}, {3268, 3265}, {5962, 68}, {6149, 255}, {6198, 1807}, {7799, 3926}, {8749, 11079}, {8882, 11077}, {8884, 1141}, {11062, 216}, {11107, 1793}

X(14165) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 11547, 2052), (468, 6530, 107)

X(14166) = PERSPECTOR OF THESE TRIANGLES: 1st MORLEY ADJUNCT AND ROUSSEL

Trilinears    4*x*y*z*(b*c*(4*x*y*z+1)+(b^2+c^2)*x+a*(b*y+c*z))-4*b*c*y^2*z^2+2*a^2*x*(y^2+z^2)-a*((-8*x^2*y*z+2*y*z-x)*(y*c+b*z)+a*y*z) : :
where x=cos(A/3), y=cos(B/3), z=cos(C/3)

Contributed by César Lozada, Aug. 26, 2017.

Let A'B'C' be the 1st Morley triangle. Let Ma be the line through A parallel to B'C', and define Mb and Mc cyclically. Let A" = Mb∩Mc, B" = Mc∩Ma, C" = Ma∩Mb. Triangle A"B"C" is perspective to the 1st adjunct Morley triangle and homothetic to the Roussel triangle at X(14166).(Randy Hutson, November 2, 2017)

X(14166) lies on these lines: {1135,10259}, {3604,8067}, {8011,11228}

X(14167) =  X(98)X(22785)∩X(262)X(6401)

Barycentrics    4*a^16*b^8 - 20*a^14*b^10 + 40*a^12*b^12 - 40*a^10*b^14 + 20*a^8*b^16 - 4*a^6*b^18 + 3*a^18*b^4*c^2 + 10*a^16*b^6*c^2 - 77*a^14*b^8*c^2 + 118*a^12*b^10*c^2 - 23*a^10*b^12*c^2 - 82*a^8*b^14*c^2 + 65*a^6*b^16*c^2 - 14*a^4*b^18*c^2 + 3*a^18*b^2*c^4 + 26*a^16*b^4*c^4 - 167*a^14*b^6*c^4 + 160*a^12*b^8*c^4 + 135*a^10*b^10*c^4 - 154*a^8*b^12*c^4 - 73*a^6*b^14*c^4 + 80*a^4*b^16*c^4 - 10*a^2*b^18*c^4 + 10*a^16*b^2*c^6 - 167*a^14*b^4*c^6 + 96*a^12*b^6*c^6 + 288*a^10*b^8*c^6 + 36*a^8*b^10*c^6 - 159*a^6*b^12*c^6 - 108*a^4*b^14*c^6 + 54*a^2*b^16*c^6 - 2*b^18*c^6 + 4*a^16*c^8 - 77*a^14*b^2*c^8 + 160*a^12*b^4*c^8 + 288*a^10*b^6*c^8 + 240*a^8*b^8*c^8 + 171*a^6*b^10*c^8 - 72*a^4*b^12*c^8 - 102*a^2*b^14*c^8 + 12*b^16*c^8 - 20*a^14*c^10 + 118*a^12*b^2*c^10 + 135*a^10*b^4*c^10 + 36*a^8*b^6*c^10 + 171*a^6*b^8*c^10 + 228*a^4*b^10*c^10 + 58*a^2*b^12*c^10 - 30*b^14*c^10 + 40*a^12*c^12 - 23*a^10*b^2*c^12 - 154*a^8*b^4*c^12 - 159*a^6*b^6*c^12 - 72*a^4*b^8*c^12 + 58*a^2*b^10*c^12 + 40*b^12*c^12 - 40*a^10*c^14 - 82*a^8*b^2*c^14 - 73*a^6*b^4*c^14 - 108*a^4*b^6*c^14 - 102*a^2*b^8*c^14 - 30*b^10*c^14 + 20*a^8*c^16 + 65*a^6*b^2*c^16 + 80*a^4*b^4*c^16 + 54*a^2*b^6*c^16 + 12*b^8*c^16 - 4*a^6*c^18 - 14*a^4*b^2*c^18 - 10*a^2*b^4*c^18 - 2*b^6*c^18 + (4*a^16*b^6 - 16*a^14*b^8 + 20*a^12*b^10 - 20*a^8*b^14 + 16*a^6*b^16 - 4*a^4*b^18 + 3*a^18*b^2*c^2 + 8*a^16*b^4*c^2 - 46*a^14*b^6*c^2 + 3*a^12*b^8*c^2 + 133*a^10*b^10*c^2 - 134*a^8*b^12*c^2 + 12*a^6*b^14*c^2 + 27*a^4*b^16*c^2 - 6*a^2*b^18*c^2 + 8*a^16*b^2*c^4 - 80*a^14*b^4*c^4 - 53*a^12*b^6*c^4 + 295*a^10*b^8*c^4 - 26*a^8*b^10*c^4 - 192*a^6*b^12*c^4 + 25*a^4*b^14*c^4 + 25*a^2*b^16*c^4 - 2*b^18*c^4 + 4*a^16*c^6 - 46*a^14*b^2*c^6 - 53*a^12*b^4*c^6 + 356*a^10*b^6*c^6 + 216*a^8*b^8*c^6 - 16*a^6*b^10*c^6 - 193*a^4*b^12*c^6 - 14*a^2*b^14*c^6 + 10*b^16*c^6 - 16*a^14*c^8 + 3*a^12*b^2*c^8 + 295*a^10*b^4*c^8 + 216*a^8*b^6*c^8 + 312*a^6*b^8*c^8 + 145*a^4*b^10*c^8 - 73*a^2*b^12*c^8 - 18*b^14*c^8 + 20*a^12*c^10 + 133*a^10*b^2*c^10 - 26*a^8*b^4*c^10 - 16*a^6*b^6*c^10 + 145*a^4*b^8*c^10 + 136*a^2*b^10*c^10 + 10*b^12*c^10 - 134*a^8*b^2*c^12 - 192*a^6*b^4*c^12 - 193*a^4*b^6*c^12 - 73*a^2*b^8*c^12 + 10*b^10*c^12 - 20*a^8*c^14 + 12*a^6*b^2*c^14 + 25*a^4*b^4*c^14 - 14*a^2*b^6*c^14 - 18*b^8*c^14 + 16*a^6*c^16 + 27*a^4*b^2*c^16 + 25*a^2*b^4*c^16 + 10*b^6*c^16 - 4*a^4*c^18 - 6*a^2*b^2*c^18 - 2*b^4*c^18)*S : :

X(14167) lies on these lines: {98, 22785}, {262, 6401}, {9755, 11984}, {9756, 11986}, {9772, 22499}, {10837, 11937}, {10838, 11938}, {10839, 11941}, {10840, 11942}, {10841, 11959}, {10842, 11960}, {10843, 11963}, {10844, 11964}, {10845, 11967}, {10846, 11969}, {10847, 11971}, {10848, 11973}, {10849, 11975}, {10850, 11977}, {10851, 11979}, {10852, 11981}, {11983, 14168}, {15182, 19390}

X(14168) =  X(98)X(22785)∩X(262)X(6401)

Barycentrics    4*a^16*b^8 - 20*a^14*b^10 + 40*a^12*b^12 - 40*a^10*b^14 + 20*a^8*b^16 - 4*a^6*b^18 + 3*a^18*b^4*c^2 + 10*a^16*b^6*c^2 - 77*a^14*b^8*c^2 + 118*a^12*b^10*c^2 - 23*a^10*b^12*c^2 - 82*a^8*b^14*c^2 + 65*a^6*b^16*c^2 - 14*a^4*b^18*c^2 + 3*a^18*b^2*c^4 + 26*a^16*b^4*c^4 - 167*a^14*b^6*c^4 + 160*a^12*b^8*c^4 + 135*a^10*b^10*c^4 - 154*a^8*b^12*c^4 - 73*a^6*b^14*c^4 + 80*a^4*b^16*c^4 - 10*a^2*b^18*c^4 + 10*a^16*b^2*c^6 - 167*a^14*b^4*c^6 + 96*a^12*b^6*c^6 + 288*a^10*b^8*c^6 + 36*a^8*b^10*c^6 - 159*a^6*b^12*c^6 - 108*a^4*b^14*c^6 + 54*a^2*b^16*c^6 - 2*b^18*c^6 + 4*a^16*c^8 - 77*a^14*b^2*c^8 + 160*a^12*b^4*c^8 + 288*a^10*b^6*c^8 + 240*a^8*b^8*c^8 + 171*a^6*b^10*c^8 - 72*a^4*b^12*c^8 - 102*a^2*b^14*c^8 + 12*b^16*c^8 - 20*a^14*c^10 + 118*a^12*b^2*c^10 + 135*a^10*b^4*c^10 + 36*a^8*b^6*c^10 + 171*a^6*b^8*c^10 + 228*a^4*b^10*c^10 + 58*a^2*b^12*c^10 - 30*b^14*c^10 + 40*a^12*c^12 - 23*a^10*b^2*c^12 - 154*a^8*b^4*c^12 - 159*a^6*b^6*c^12 - 72*a^4*b^8*c^12 + 58*a^2*b^10*c^12 + 40*b^12*c^12 - 40*a^10*c^14 - 82*a^8*b^2*c^14 - 73*a^6*b^4*c^14 - 108*a^4*b^6*c^14 - 102*a^2*b^8*c^14 - 30*b^10*c^14 + 20*a^8*c^16 + 65*a^6*b^2*c^16 + 80*a^4*b^4*c^16 + 54*a^2*b^6*c^16 + 12*b^8*c^16 - 4*a^6*c^18 - 14*a^4*b^2*c^18 - 10*a^2*b^4*c^18 - 2*b^6*c^18 - (4*a^16*b^6 - 16*a^14*b^8 + 20*a^12*b^10 - 20*a^8*b^14 + 16*a^6*b^16 - 4*a^4*b^18 + 3*a^18*b^2*c^2 + 8*a^16*b^4*c^2 - 46*a^14*b^6*c^2 + 3*a^12*b^8*c^2 + 133*a^10*b^10*c^2 - 134*a^8*b^12*c^2 + 12*a^6*b^14*c^2 + 27*a^4*b^16*c^2 - 6*a^2*b^18*c^2 + 8*a^16*b^2*c^4 - 80*a^14*b^4*c^4 - 53*a^12*b^6*c^4 + 295*a^10*b^8*c^4 - 26*a^8*b^10*c^4 - 192*a^6*b^12*c^4 + 25*a^4*b^14*c^4 + 25*a^2*b^16*c^4 - 2*b^18*c^4 + 4*a^16*c^6 - 46*a^14*b^2*c^6 - 53*a^12*b^4*c^6 + 356*a^10*b^6*c^6 + 216*a^8*b^8*c^6 - 16*a^6*b^10*c^6 - 193*a^4*b^12*c^6 - 14*a^2*b^14*c^6 + 10*b^16*c^6 - 16*a^14*c^8 + 3*a^12*b^2*c^8 + 295*a^10*b^4*c^8 + 216*a^8*b^6*c^8 + 312*a^6*b^8*c^8 + 145*a^4*b^10*c^8 - 73*a^2*b^12*c^8 - 18*b^14*c^8 + 20*a^12*c^10 + 133*a^10*b^2*c^10 - 26*a^8*b^4*c^10 - 16*a^6*b^6*c^10 + 145*a^4*b^8*c^10 + 136*a^2*b^10*c^10 + 10*b^12*c^10 - 134*a^8*b^2*c^12 - 192*a^6*b^4*c^12 - 193*a^4*b^6*c^12 - 73*a^2*b^8*c^12 + 10*b^10*c^12 - 20*a^8*c^14 + 12*a^6*b^2*c^14 + 25*a^4*b^4*c^14 - 14*a^2*b^6*c^14 - 18*b^8*c^14 + 16*a^6*c^16 + 27*a^4*b^2*c^16 + 25*a^2*b^4*c^16 + 10*b^6*c^16 - 4*a^4*c^18 - 6*a^2*b^2*c^18 - 2*b^4*c^18)*S ::

X(14168) lies on these lines: {98, 22786}, {262, 6402}, {9755, 11985}, {9756, 11987}, {9772, 22500}, {10837, 11939}, {10838, 11940}, {10839, 11943}, {10840, 11944}, {10841, 11961}, {10842, 11962}, {10843, 11965}, {10844, 11966}, {10845, 11970}, {10846, 11968}, {10847, 11974}, {10848, 11972}, {10849, 11978}, {10850, 11976}, {10851, 11982}, {10852, 11980}, {11983, 14167}, {15182, 19391}

leftri

Le Viet An equilateral triangles: X(14169)-X(14188)

rightri

This preamble and centers X(14169)-X(14188) were contributed by César Eliud Lozada, August 30, 2017.

Let ABC be a triangle and BCA', CAB', ABC' equilateral triangles erected out/in - wardly of ABC. Let Bc, Cb be the circumcenters of CC'B and BB'C, respectively, and build Ca, Ac, Ab and Ba cyclically. Denote the circumcenters of BcCbA, CaAcB, AbBaC as Oa, Ob, Oc, respectively. Then, in each case, the triangle OaObOc is equilateral. (See: Hyacinthos 26551).

For BCA', CAB', ABC' built outwards ABC, the triangle OaObOc will be referred here as the outer-Le Viet An triangle.
This triangle has sidelength ObOc = 4*S^2*R*|(SW+sqrt(3)*S)/((sqrt (3)*SA+S)*(sqrt(3)*SB+S)*(sqrt (3)*SC+S))|.
Oa has coordinates: Oa = (SW+sqrt(3)*S)*a : (SB-SC)*b : (SC-SB)*c (trilinears)

For BCA', CAB', ABC' built inwards ABC, the triangle OaObOc will be referred here as the inner-Le Viet An triangle.
This triangle has sidelength ObOc = 4*S^2*R*|(SW-sqrt(3)*S)/((sqrt (3)*SA-S)*(sqrt(3)*SB-S)*(sqrt (3)*SC-S))|
Oa has coordinates: Oa = (SW-sqrt(3)*S)*a : (SB-SC)*b : (SC-SB)*c (trilinears)

Both triangles are perspective to the anti-orthocentroidal triangle.

The outer-Le Viet An triangle is orthologic to these triangles: 1st Ehrmann, inner-Le Viet An and inner-Napoleon. It is also parallelogic to these triangles: inner-Le Viet An and outer-Napoleon.

The inner-Le Viet An triangle is orthologic to these triangles: 1st Ehrmann, outer-Le Viet An, 1st Morley, 2nd Morley, 3rd Morley, outer-Napoleon, Roussel, Stammler. It is also parallelogic to these triangles: 1st Ehrmann, outer-Le Viet An, 1st Morley, 2nd Morley, 3rd Morley, outer-Napoleon, Roussel, Stammler.

When constructing the outer- and the inner- Le Viet An triangles, the Euler lines of BcCbA, CaAcB, AbBaC are concurrent at X(14185) and X(14187), respectively.


X(14169) = CENTER OF THE INNER-LE VIET AN TRIANGLE

Trilinears    (3*a^4-2*(b^2+c^2)*a^2-b^4-c^4+2*sqrt(3)*(b^2+c^2-a^2)*S)*a : :

X(14169) lies on these lines: {2,9750}, {3,74}, {4,8836}, {15,11003}, {16,23}, {62,3458}, {184,9735}, {511,11126}, {1495,13349}, {2782,14181}, {2854,14179}, {3129,5640}, {3131,5012}, {3171,5238}, {5464,9143}, {5611,11422}, {6773,11092}, {7712,10646}, {11141,11626}, {14176,14184}

X(14169) = midpoint of X(i) and X(j) for these {i,j}: {14176, 14184}, {14181, 14187}
X(14169) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 110, 11130), (3, 6800, 14170)

X(14170) = CENTER OF THE OUTER-LE VIET AN TRIANGLE

Trilinears    (3*a^4-2*(b^2+c^2)*a^2-b^4-c^4-2*sqrt(3)*(b^2+c^2-a^2)*S)*a : :

X(14170) lies on these lines: {2,9749}, {3,74}, {4,8838}, {15,23}, {16,11003}, {61,3457}, {184,9736}, {511,11127}, {1495,13350}, {2782,14177}, {2854,14173}, {3130,5640}, {3132,5012}, {3170,5237}, {5463,9143}, {5615,11422}, {6770,11078}, {7712,10645}, {11142,11624}, {14175,14183}

X(14170) = midpoint of X(i) and X(j) for these {i,j}: {14175, 14183}, {14177, 14185}
X(14170) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 110, 11131), (3, 6800, 14169)

X(14171) = PERSPECTOR OF THIS TRIANGLES: INNER-LE VIET AN AND ANTI-ORTHOCENTROIDAL

Trilinears
(-6*sqrt(3)*(a^10-(b^2+c^2)*a^8-(2*b^2-c^2)*(b^2-2*c^2)*a^6+2*(b^4-c^4)*(b^2-c^2)*a^4+(b^2-c^2)^2*(b^4+c^4)*a^2-(b^4-c^4)*(b^2-c^2)*((b^2+c^2)^2-b^2*c^2))*S+a^12+4*(b^2+c^2)*a^10-(b^4+19*b^2*c^2+c^4)*a^8-(b^2+c^2)*(32*b^4-71*b^2*c^2+32*c^4)*a^6+(47*b^8+47*c^8-24*(b^2+c^2)^2*b^2*c^2)*a^4-(b^4-c^4)*(b^2-c^2)*(20*b^4+21*b^2*c^2+20*c^4)*a^2+(b^4+5*b^2*c^2+c^4)*(b^2-c^2)^4)*a : :

X(14171) lies on the line {5663,13858}

X(14172) = PERSPECTOR OF THIS TRIANGLES: OUTER-LE VIET AN AND ANTI-ORTHOCENTROIDAL

Trilinears
(6*sqrt(3)*(a^10-(b^2+c^2)*a^8-(2*b^2-c^2)*(b^2-2*c^2)*a^6+2*(b^4-c^4)*(b^2-c^2)*a^4+(b^2-c^2)^2*(b^4+c^4)*a^2-(b^4-c^4)*(b^2-c^2)*((b^2+c^2)^2-b^2*c^2))*S+a^12+4*(b^2+c^2)*a^10-(b^4+19*b^2*c^2+c^4)*a^8-(b^2+c^2)*(32*b^4-71*b^2*c^2+32*c^4)*a^6+(47*b^8+47*c^8-24*(b^2+c^2)^2*b^2*c^2)*a^4-(b^4-c^4)*(b^2-c^2)*(20*b^4+21*b^2*c^2+20*c^4)*a^2+(b^4+5*b^2*c^2+c^4)*(b^2-c^2)^4)*a : :

X(14172) lies on the line {5663,13859}

X(14173) = ORTHOLOGIC CENTER OF THESE TRIANGLES: OUTER-LE VIET AN TO 1st EHRMANN

Trilinears    (2*(2*a^6+2*(b^2+c^2)*a^4-(2*b^4+7*b^2*c^2+2*c^4)*a^2-2*b^6-2*c^6)*S+sqrt(3)*(2*(b^2+c^2)*a^2-2*b^4-3*b^2*c^2-2*c^4)*(a^2+b^2+c^2)*a^2)*a : :

X(14173) lies on these lines: {3,67}, {6,3130}, {15,12367}, {22,5859}, {61,9971}, {2854,14170}, {9019,11127}, {9149,14177}

X(14173) = {X(2930), X(7669)}-harmonic conjugate of X(14179)

X(14174) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st EHRMANN TO OUTER-LE VIET AN

Barycentrics    (3*(3*S^2+3*SA^2-2*(9*R^2-SW)*SA)*S^2-sqrt(3)*S*((18*R^2-7*SW)*S^2-3*SW*SA^2)+2*SW^3*SA)*(SB+SC) : :

X(14174) lies on these lines: {15,111}, {531,11632}, {10204,14180}

X(14175) = ORTHOLOGIC CENTER OF THESE TRIANGLES: OUTER-LE VIET AN TO INNER-LE VIET AN

Trilinears
(-2*sqrt(3)*(2*a^6-(b^2+c^2)*a^4+(b^4-4*b^2*c^2+c^4)*a^2-(2*b^2-c^2)*(b^2-2*c^2)*(b^2+c^2))*S+5*(b^2+c^2)*a^6-2*(5*b^4-b^2*c^2+5*c^4)*a^4+(b^2+c^2)*(7*b^4-11*b^2*c^2+7*c^4)*a^2-2*b^8-2*c^8-b^2*c^2*(7*b^4-20*b^2*c^2+7*c^4))*a : :

X(14175) lies on these lines: {16,110}, {530,14185}, {14170,14183}

X(14175) = reflection of X(14183) in X(14170)

X(14176) = ORTHOLOGIC CENTER OF THESE TRIANGLES: INNER-LE VIET AN TO OUTER-LE VIET AN

Trilinears
(2*sqrt(3)*(2*a^6-(b^2+c^2)*a^4+(b^4-4*b^2*c^2+c^4)*a^2-(2*b^2-c^2)*(b^2-2*c^2)*(b^2+c^2))*S+5*(b^2+c^2)*a^6-2*(5*b^4-b^2*c^2+5*c^4)*a^4+(b^2+c^2)*(7*b^4-11*b^2*c^2+7*c^4)*a^2-2*b^8-2*c^8-b^2*c^2*(7*b^4-20*b^2*c^2+7*c^4))*a : :

X(14176) lies on these lines: {15,110}, {531,14187}, {14169,14184}

X(14176) = reflection of X(14184) in X(14169)

X(14177) = ORTHOLOGIC CENTER OF THESE TRIANGLES: OUTER-LE VIET AN TO INNER-NAPOLEON

Barycentrics    -2*sqrt(3)*a^2*(a^2-b^2)*(a^2-c^2)*S+3*a^8-2*(b^2+c^2)*a^6+(b^4-3*b^2*c^2+c^4)*a^4-(2*b^2-c^2)*(b^2-2*c^2)*(b^2+c^2)*a^2-2*(b^2-c^2)^2*b^2*c^2 : :

X(14177) lies on these lines: {2,98}, {530,14183}, {2782,14170}, {6033,8838}, {6770,6773}, {9149,14173}

X(14177) = reflection of X(14185) in X(14170)

X(14178) = ORTHOLOGIC CENTER OF THESE TRIANGLES: INNER-NAPOLEON TO OUTER-LE VIET AN

Barycentrics    (S^2*(2*S^2-3*R^2*(3*SA-SW)+SA*(SW+3*SA)-SW^2)-sqrt(3)*((3*R^2-SW)*S^2-(3*R^2+SA)*SA*SW)*S+SA*SW^3)*(SB+SC) : :

X(14178) lies on the Napoleon-inner circle and these lines: {2,14186}, {13,511}, {15,3231}, {512,5463}, {3642,7998}, {3643,6787}

X(14178) = reflection of X(14186) in X(2)
X(14178) = anticomplement of X(33480)
X(14178) = antipode of X(14186) in Napoleon-inner circle

X(14179) = ORTHOLOGIC CENTER OF THESE TRIANGLES: INNER-LE VIET AN TO 1st EHRMANN

Trilinears    (-2*(2*a^6+2*(b^2+c^2)*a^4-(2*b^4+7*b^2*c^2+2*c^4)*a^2-2*b^6-2*c^6)*S+sqrt(3)*(2*(b^2+c^2)*a^2-2*b^4-3*b^2*c^2-2*c^4)*(a^2+b^2+c^2)*a^2)*a : :

X(14179) lies on these lines: {3,67}, {6,3129}, {16,12367}, {22,5858}, {62,9971}, {2854,14169}, {9019,11126}, {9149,14181}

X(14179) = {X(2930), X(7669)}-harmonic conjugate of X(14173)

X(14180) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st EHRMANN TO INNER-LE VIET AN

Barycentrics    (3*(3*S^2+3*SA^2-2*(9*R^2-SW)*SA)*S^2+sqrt(3)*S*((18*R^2-7*SW)*S^2-3*SW*SA^2)+2*SW^3*SA)*(SB+SC) : :

X(14180) lies on these lines: {16,111}, {530,11632}, {10204,14174}

X(14181) = ORTHOLOGIC CENTER OF THESE TRIANGLES: INNER-LE VIET AN TO OUTER-NAPOLEON

Barycentrics    -2*sqrt(3)*a^2*(a^2-b^2)*(c^2-a^2)*S+3*a^8-2*(b^2+c^2)*a^6+(b^4-3*b^2*c^2+c^4)*a^4-(2*b^2-c^2)*(b^2-2*c^2)*(b^2+c^2)*a^2-2*(b^2-c^2)^2*b^2*c^2 : :

X(14181) lies on these lines: {2,98}, {531,14184}, {2782,14169}, {6033,8836}, {6770,6773}, {9149,14179}, {11130,12042}

X(14181) = reflection of X(14187) in X(14169)

X(14182) = ORTHOLOGIC CENTER OF THESE TRIANGLES: OUTER-NAPOLEON TO INNER-LE VIET AN

Barycentrics    (S^2*(2*S^2-3*R^2*(3*SA-SW)+SA*(SW+3*SA)-SW^2)+sqrt(3)*((3*R^2-SW)*S^2-(3*R^2+SA)*SA*SW)*S+SA*SW^3)*(SB+SC) : :

X(14182) lies on the Napoleon-outer circle and these lines: {2,14188}, {14,511}, {16,3231}, {512,5464}, {3642,6787}, {3643,7998}

X(14182) = reflection of X(14188) in X(2)
X(14182) = anticomplement of X(33481)
X(14182) = antipode of X(14188) in Napoleon-outer circle

X(14183) = PARALLELOGIC CENTER OF THESE TRIANGLES: OUTER-LE VIET AN TO INNER-LE VIET AN

Trilinears    a*(2*(2*a^2-b^2-c^2)*S-((b^2+c^2)*a^2+b^4-4*b^2*c^2+c^4)*sqrt(3))*(a^2-b^2)*(a^2-c^2) : :

X(14183) lies on these lines: {16,1338}, {22,11630}, {110,9163}, {530,14177}, {691,5467}, {805,5995}, {10410,10411}, {14170,14175}

X(14183) = reflection of X(14175) in X(14170)
X(14183) = {X(5467), X(11634)}-harmonic conjugate of X(14184)

X(14184) = PARALLELOGIC CENTER OF THESE TRIANGLES: INNER-LE VIET AN TO OUTER-LE VIET AN

Trilinears    a*(a^2-b^2)*(a^2-c^2)*(2*(2*a^2-b^2-c^2)*S+((b^2+c^2)*a^2+b^4-4*b^2*c^2+c^4)*sqrt(3)) : :

X(14184) lies on these lines: {15,1337}, {22,11629}, {110,9162}, {531,14181}, {691,5467}, {805,5994}, {10409,10411}, {14169,14176}

X(14184) = reflection of X(14176) in X(14169)

X(14185) = PARALLELOGIC CENTER OF THESE TRIANGLES: OUTER-LE VIET AN TO INNER-NAPOLEON

Barycentrics    (-2*S*sqrt(3)*a^2+3*a^4-3*(b^2+c^2)*a^2+2*(b^2-c^2)^2)*(a^2-b^2)*(a^2-c^2) : :

X(14185) lies on these lines: {2,9735}, {99,110}, {476,9202}, {530,14175}, {1316,11130}, {2782,14170}, {6321,8838}

X(14185) = reflection of X(14177) in X(14170)

X(14186) = PARALLELOGIC CENTER OF THESE TRIANGLES: INNER-NAPOLEON TO OUTER-LE VIET AN

Barycentrics    ((S^2+3*R^2*(-SW+3*SA)-SW*(SA-2*SW))*S^2+sqrt(3)*S*((3*R^2+SW)*S^2-(3*R^2-SA)*SA*SW)-(SB+SC)*SA*SW^2)*(SB+SC) : :

X(14186) lies on the Napoleon-inner circle and these lines: {2,14178}, {13,512}, {15,237}, {511,5463}, {1634,5611}, {5476,14188}

X(14186) = reflection of X(14178) in X(2)
X(14186) = anticomplement of X(33490)
X(14186) = antipode of X(14178) in Napoleon-inner circle

X(14187) = PARALLELOGIC CENTER OF THESE TRIANGLES: INNER-LE VIET AN TO OUTER-NAPOLEON

Barycentrics    (2*sqrt(3)*a^2*S+3*a^4-3*(b^2+c^2)*a^2+2*(b^2-c^2)^2)*(a^2-b^2)*(c^2-a^2) : :

X(14187) lies on these lines: {2,9736}, {99,110}, {476,9203}, {531,14176}, {1316,11131}, {2782,14169}, {6321,8836}

X(14187) = reflection of X(14181) in X(14169)

X(14188) = PARALLELOGIC CENTER OF THESE TRIANGLES: OUTER-NAPOLEON TO INNER-LE VIET AN

Barycentrics    ((S^2+3*R^2*(-SW+3*SA)-SW*(SA-2*SW))*S^2-sqrt(3)*S*((3*R^2+SW)*S^2-(3*R^2-SA)*SA*SW)-(SB+SC)*SA*SW^2)*(SB+SC) : :

X(14188) lies on the Napoleon-outer circle and these lines: {2,14182}, {14,512}, {16,237}, {511,5464}, {1634,5615}, {5476,14186}

X(14188) = reflection of X(14182) in X(2)
X(14188) = anticomplement of X(33491)
X(14188) = antipode of X(14182) in Napoleon-outer circle

X(14189) = K040 IMAGE OF X(934)

Barycentrics    (a+b-c) (a-b+c) (a^4-2 a^3 b+a^2 b^2-2 a^3 c+a^2 b c+b^3 c+a^2 c^2-2 b^2 c^2+b c^3) : :
X(14189) = X(7) - 4 X(1323)

See Bernard Gibert, Pelletier strophoid (the cubic K040) and Points on the cubic K040

X(14189) lies on the cubics K040 and K294 and these lines: {1,7}, {2,2898}, {9,3177}, {55,1088}, {85,1001}, {105,927}, {241,673}, {242,273}, {243,13149}, {348,2550}, {479,9778}, {514,657}, {518,664}, {658,1155}, {934,2724}, {1308,2369}, {1441,7112}, {1445,2082}, {1996,5218}, {2377,2737}, {2651,4573}, {3474,7056}, {3757,6063}, {4554,5205}, {5572,10509}, {5853,9436}

X(14189) = crosspoint of X(9442) and X(9445)
X(14189) = crossdifference of every pair of points on line {657, 2293}
X(14189) = crosssum of X(i) and X(j) for these (i,j): {926, 3022}, {9440, 9441}
X(14189) = Adams-circle inverse of X(7)
X(14189) = X(99)-beth conjugate of X(518)
X(14189) = X(9442)-zayin conjugate of X(672)
X(14189) = X(i)-Ceva conjugate of X(j) for these (i,j): {241, 1447}, {673, 7}, {9442, 9446}
X(14189) = X(9441)-cross conjugate of X(10025)
X(14189) = isoconjugate of X(55) and X(9442)
X(14189) = X(1)-Hirst inverse of X(7)
X(14189) = X(1)-line conjugate of X(2293)
X(14189) = X(2076)-of-intouch-triangle
X(14189) = perspector of ABC and reflection of intouch triangle in Gergonne line
X(14189) = Gibert-Burek-Moses concurrent circles image of X(170)
X(14189) = perspector of conic {A,B,C,PU(94)}
X(14189) = crossdifference of PU(104)
X(14189) = barycentric product X(i)*X(j) for these {i,j}: {7, 10025}, {85, 9441}
X(14189) = barycentric quotient X(i)/X(j) for these {i,j}: {57, 9442}, {9441, 9}, {10025, 8}
X(14189) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (55, 1088, 9446), (279, 390, 7), (3160, 3188, 7176), (3674, 12573, 7)

X(14190) = K040 IMAGE OF X(106)

Barycentrics    a (a+b-2 c) (a-2 b+c) (2 a^3-2 a^2 b+a b^2-b^3-2 a^2 c+b^2 c+a c^2+b c^2-c^3) : :
X(14190) = 3 X(238) - X(3245) = 3 X(1168) - X(4792) =3 X(1279) - 2 X(5126)

See Bernard Gibert, Pelletier strophoid (the cubic K040) and Points on the cubic K040

X(14190) lies on the cubics K040 and K716 and these lines: {1,513}, {44,517}, {88,105}, {106,1279}, {238,3245}, {243,6336}, {518,1156}, {679,1318}, {4080,5057}, {4555,4702}, {4997,5087}, {5048,9326}

X(14190) = midpoint of X(1320) and X(3257)
X(14190) = reflection of X(1155) in X(3246)
X(14190) = {X(99),X(901)}-harmonic conjugate of X(1155)
X(14190) = X(2246)-zayin conjugate of X(44)
X(14190) = isoconjugate of X(j) and X(j) for these (i,j): {519, 840}
X(14190) = X(517)-line conjugate of X(44)
X(14190) = X(36)-vertex conjugate of X(1022)
X(14190) = trilinear pole of line {1643, 2246}
X(14190) = barycentric product X(i)*X(j) for these {i,j}: {88, 528}, {903, 2246}, {1320, 5723}, {1643, 4555}
X(14190) = barycentric quotient X(i)/X(j) for these {i,j}: {528, 4358}, {1643, 900}, {2246, 519}, {9456, 840}

X(14191) = K040 IMAGE OF X(840)

Barycentrics    a (2 a-b-c) (a^3-a^2 b-a b^2+b^3-a^2 c-b^2 c+2 a c^2+2 b c^2-2 c^3) (a^3-a^2 b+2 a b^2-2 b^3-a^2 c+2 b^2 c-a c^2-b c^2+c^3) : :

See Bernard Gibert, Pelletier strophoid (the cubic K040) and Points on the cubic K040

X(14191) lies on the cubic K040 and these lines: {1,2254}, {100,518}, {105,1156}, {214,1023}, {1320,4618}, {2246,2801}, {4152,6174}, {4597,10031}, {4712,6594}, {5528,9451}

X(14191) = reflection of X(1023) in X(214)
X(14191) = crossdifference of every pair of points on line {1643, 2246}
X(14191) = X(44)-zayin conjugate of X(2246)
X(14191) = isoconjugate of X(j) and X(j) for these (i,j): {88, 2246}, {106, 528}, {1643, 3257}, {2316, 5723}
X(14191) = X(2801)-line conjugate of X(2246)
X(14191) = barycentric product X(840)*X(4358)
X(14191) = barycentric quotient X(i)/X(j) for these {i,j}: {44, 528}, {840, 88}, {902, 2246}, {1319, 5723}, {1960, 1643}

X(14192) = K040 IMAGE OF X(112)

Barycentrics    a (a+b) (a+c) (a^2+b^2-c^2) (a^2-b^2+c^2) (a^5-a^3 b^2-a^2 b^3+b^5-a^3 b c+a b^3 c-a^3 c^2+2 a b^2 c^2-b^3 c^2-a^2 c^3+a b c^3-b^2 c^3+c^5) : :

X(14192) lies on the cubic K040 and these lines: {1,19}, {29,6284}, {107,243}, {162,1155}, {250,1325}, {415,5993}, {2074,10058}, {2075,5172}, {5217,11107}

X(14192) = X(1)-Hirst inverse of X(1172)

X(14193) = K040 IMAGE OF X(901)

Barycentrics    a (a+b-2 c) (a-2 b+c) (3 a^3-4 a^2 b+b^3-4 a^2 c+7 a b c-2 b^2 c-2 b c^2+c^3) : :
X(100) + 2 X(1054), 4 X(106) - X(1320)

See Bernard Gibert, Pelletier strophoid (the cubic K040) and Points on the cubic K040

X(14193) lies on the cubic K040 and these lines: {1,88}, {903,6174}, {1155,3257}, {1635,2827}, {2796,4945}, {3035,4997}, {4582,5205}

X(14193) = X(1)-Hirst inverse of X(1320)
X(14193) = {X(2827),X(12034)}-line conjugate of X(1635)
X(14193) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (88, 100, 1320), (106, 1054, 88), (214, 9324, 100)

X(14194) = K040 IMAGE OF X(759)

Barycentrics    (a+b) (a+c) (a^2-a b+b^2-c^2) (a^2-b^2-a c+c^2) (a^5 b-a^4 b^2-a^3 b^3+a^2 b^4+a^5 c-b^5 c-a^4 c^2-a^3 c^3+2 b^3 c^3+a^2 c^4-b c^5) : :

X(14194) lies on the cubic K040 and these lines: 1,523}, {30,2245}, {105,476}, {243,2073}, {518,6740}, {759,2690}, {1155,5196}

X(14194) = X(30)-line conjugate of X(2245)

X(14195) = K040 IMAGE OF X(99)

Barycentrics    b c (a+b) (a+c) (a^4 b-a^3 b^2+a^4 c-a^3 b c+a^2 b^2 c-a^3 c^2+a^2 b c^2+a b^2 c^2-b^3 c^2-b^2 c^3) : :

See Bernard Gibert, Pelletier strophoid (the cubic K040) and Points on the cubic K040

X(14195) lies on the cubic K040 and these lines: {1,75}, {99,1921}, {243,6331}, {518,7257}, {798,6002}, {799,1155}, {2651,4631}

X(14195) = X(1)-Hirst inverse of X(314)
X(14195) = X(6002)-line conjugate of X(798)

X(14196) = K040 IMAGE OF X(741)

Barycentrics    a^2 (a+b) (a+c) (-b^2+a c) (a b-c^2) (a^3 b-b^4+a^3 c-2 a^2 b c-a b^2 c+b^3 c-a b c^2+2 b^2 c^2+b c^3-c^4) : :

See Bernard Gibert, Pelletier strophoid (the cubic K040) and Points on the cubic K040

X(14196) lies on the cubic K040 and these lines: {1,512}, {105,805}, {511,2238}, {518,7061}, {741,1326}, {4589,9470}

X(14196) = X(511)-line conjugate of X(2238)

X(14197) = K040 IMAGE OF X(105)

Barycentrics    (a^2+b^2-a c-b c) (a^2-a b-b c+c^2) (a^4 b-2 a^3 b^2+a^2 b^3+a^4 c-b^4 c-2 a^3 c^2+b^3 c^2+a^2 c^3+b^2 c^3-b c^4) : :

See Bernard Gibert, Pelletier strophoid (the cubic K040) and Points on the cubic K040

X(14197) lies on the cubic K040 and these lines: {1,514}, {105,927}, {242,5089}, {516,672}, {518,10025}, {673,1155}

X(14197) = X(927)-beth conjugate of X(1456)
X(14197) = isoconjugate of X(j) and X(j) for these (i,j): {518, 12032}
X(14197) = X(1)-Hirst inverse of X(885)
X(14197) = X(516)-line conjugate of X(672)
X(14197) = barycentric quotient X(i)/X(j) for these {i,j}: {1438, 12032}

X(14198) = K040 IMAGE OF X(104)

Barycentrics    (a^3-a^2 b-a b^2+b^3+2 a b c-a c^2-b c^2) (a^3-a b^2-a^2 c+2 a b c-b^2 c-a c^2+c^3) (a^5 b-a^4 b^2-a^3 b^3+a^2 b^4+a^5 c-2 a^4 b c+2 a^3 b^2 c-a^2 b^3 c+a b^4 c-b^5 c-a^4 c^2+2 a^3 b c^2-a b^3 c^2-a^3 c^3-a^2 b c^3-a b^2 c^3+2 b^3 c^3+a^2 c^4+a b c^4-b c^5) : :

See Bernard Gibert, Pelletier strophoid (the cubic K040) and Points on the cubic K040

X(14198) lies on the cubic K040 and these lines: {1,522}, {104,929}, {105,243}, {296,518}, {515,2183}, {3685,13136}

X(14198) = X(515)-line conjugate of X(2183)

X(14199) = K040 IMAGE OF X(932)

Barycentrics    (a^2-b c) (a b-a c-b c) (a b-a c+b c) (a^2 b-a b^2+a^2 c-a b c+b^2 c-a c^2+b c^2) : :

See Bernard Gibert, Pelletier strophoid (the cubic K040) and Points on the cubic K040

X(14199) lies on the cubics K040 and K862 and these lines: {1,87}, {75,2053}, {242,1281}, {1155,4598}, {7061,8858}

X(14199) = Neuberg-circles-radical-circle-inverse of X(1)
X(14199) = X(1)-Hirst inverse of X(7155)

X(14200) = K040 IMAGE OF X(813)

Barycentrics    a (-b^2+a c) (a b-c^2) (a^4-2 a^2 b^2+a b^3-a^2 b c+2 a b^2 c-2 a^2 c^2+2 a b c^2-2 b^2 c^2+a c^3) : :

See Bernard Gibert, Pelletier strophoid (the cubic K040) and Points on the cubic K040

X(14200) lies on the cubic K040 and these lines: {1,39}, {100,7077}, {190,1281}, {660,1155}, {813,9470}

X(14200) = X(1)-Hirst inverse of X(4876)

X(14201) = K040 IMAGE OF X(1477)

Barycentrics    a (a^2-a b+2 b^2-2 a c-b c+c^2) (a^2-2 a b+b^2-a c-b c+2 c^2) (2 a^4-2 a^3 b+a^2 b^2-2 a b^3+b^4-2 a^3 c+2 a b^2 c-2 b^3 c+a^2 c^2+2 a b c^2+2 b^2 c^2-2 a c^3-2 b c^3+c^4) : :

See Bernard Gibert, Pelletier strophoid (the cubic K040) and Points on the cubic K040

X(14201) lies on the cubic K040 and these lines: {1,3309}, {105,518}, {1477,2736}, {2496,9318}

X(14201) = midpoint of X(1280) and X(6078)
X(14201) = X(105)-line conjugate of X(2348)

X(14202) = K040 IMAGE OF X(111)

Barycentrics    a (a^2+b^2-2 c^2) (a^2-2 b^2+c^2) (2 a^4-a^3 b-2 a^2 b^2+2 a b^3-b^4-a^3 c+2 a^2 b c-a b^2 c-b^3 c-2 a^2 c^2-a b c^2+4 b^2 c^2+2 a c^3-b c^3-c^4) : :

See Bernard Gibert, Pelletier strophoid (the cubic K040) and Points on the cubic K040

X(14202) lies on the cubic K040 and these lines: {1,661}, {895,8540}, {897,1155}, {1156,2651}

X(14203) = K040 IMAGE OF X(102)

Barycentrics    a (a^4-2 a^2 b^2+b^4-a^3 c+a^2 b c+a b^2 c-b^3 c+a^2 c^2-2 a b c^2+b^2 c^2+a c^3+b c^3-2 c^4) (a^4-a^3 b+a^2 b^2+a b^3-2 b^4+a^2 b c-2 a b^2 c+b^3 c-2 a^2 c^2+a b c^2+b^2 c^2-b c^3+c^4) (2 a^6-2 a^5 b-a^4 b^2+2 a b^5-b^6-2 a^5 c+4 a^4 b c-2 a b^4 c-a^4 c^2+b^4 c^2-2 a b c^4+b^2 c^4+2 a c^5-c^6) : :

See Bernard Gibert, Pelletier strophoid (the cubic K040) and Points on the cubic K040

X(14203) lies on the cubic K040 and these lines: {1,521}, {102,2728}, {105,6081}, {2182,6001}

X(14203) = X(6001)-line conjugate of X(2182)

X(14204) = K040 IMAGE OF X(2222)

Barycentrics    (a^2-a b+b^2-c^2) (a^2-b^2-a c+c^2) (a^5-a^4 b-a^3 b^2+a^2 b^3-a^4 c+3 a^3 b c-a^2 b^2 c-2 a b^3 c+b^4 c-a^3 c^2-a^2 b c^2+4 a b^2 c^2-b^3 c^2+a^2 c^3-2 a b c^3-b^2 c^3+b c^4) : :

See Bernard Gibert, Pelletier strophoid (the cubic K040) and Points on the cubic K040

X(14204) lies on the cubic K040 and these lines: {1,5}, {654,900}, {655,1155}

X(14204) = X(1)-Hirst inverse of X(80)
X(14204) = X(900)-line conjugate of X(654)
X(14204) = {X(80),X(2006)}-harmonic conjugate of X(11)

X(14205) = X(1)X(3309)∩X(3)X(884)

Barycentrics    a (b-c) (2 a^5 b-5 a^4 b^2+4 a^3 b^3-2 a^2 b^4+2 a b^5-b^6+2 a^5 c-4 a^4 b c+4 a^3 b^2 c-2 a^2 b^3 c-2 a b^4 c+2 b^5 c-5 a^4 c^2+4 a^3 b c^2+2 a^2 b^2 c^2-b^4 c^2+4 a^3 c^3-2 a^2 b c^3-2 a^2 c^4-2 a b c^4-b^2 c^4+2 a c^5+2 b c^5-c^6) : :

X(14205) is the point denoted by S in connection with the cubic K040. See Bernard Gibert, Pelletier strophoid (the cubic K040).

X(14205) lies on these lines: {1, 3309}, {3, 884}, {1643, 9441}, {2820, 3960}

leftri

Nguyen Images: X(14206) - X(14213)

rightri

Let ABC be a triangle, let L be the line through B orthogonal to BC, let BA be any point on L, and let BC be the point such that BBABCC is a rectangle. Likewise, let CCBCAA be a rectangle with base CA and let AACABB be a rectangle with base AB. Let A' be the midpoint of BACA, and define B' and C' cyclically. Let LA be the line through A' perpendicular to BC, and define LB and LC cyclically. Nguyen Ngoc Giang found that the lines LA, LB, LC concur. (Received from Dao Thanh Oai, August 11, 2017)

If you have The Geometer's Sketchpad, you can view Nguyen Rectangles.

Peter Moses analyzed the construction (August, 2017) as follows. Let U be the ratio of the height of the rectangle BBABCC to the base; that is, U = |BAB|/a. Define V and W cyclically. The point X' of concurrence is given by

X' = (a2 + b2 - c2)(a2 - b2 + c2) - 2a2SU + (a2 + b2 - c2)SV + (a2 - b2 + c2)SW : : (barycentrics)

The triangles ABC and A'B'C' are orthologic, and X' is the A'B'C'-to-ABC orthologic center. The ABC-to-A'B'C' orthologic center, X'', is given by

X'' = 1/(-a2 + b2 + c2 + SU) : :

If X is a point in the plane of ABC, then it has actual trilinear distances (possibly nonpositive) that are the heights of rectangles as in the above construction. Therefore, starting with X = x : y : z (barycentrics), we have U = kx/a2, where k = S/(x + y + z), and V = ky/b2 and W = kz/c2. Consequently,

X' = 2a(abc)2 (cos B cos C)(x + y + z) + S2 (-abcx + a2 cy cos C + a2 bz cos B) : :

Equivalently,

X' = X' = (H - abcS2)x + (H + S2 a2 c cos C)y + (H + S2 a2 b cos B)z : : , where H = 2a(abc)2 (cos B cos C).

Also,

X' = a (c2 SC (3 S2 - 2 SC SA) y + b^2 SB (3 S2 - 2 SA SB) z - b2 c2 (S^2 - 2 SB SC) x) : :

The point X' is here named the Nguyen image of X, and the point X'', the adjoint Nguyen image of X, given by

X'' = (x + y + z)/(a2SA(x + y + z) + 4S2x)

The appearance of (i,j) in the following list means that the X(j) = Nguyen image of X(i): (3,14211), (6,92), (511,14206), (512,14207), (513,4468), (518,908), (520,14208), (523,661), (524,1959), (525,14209), (1350,14212), (1351,14213), (2574,2582), (2575,2583), (8675,1577), (8999,4391), (9000,693), (9001,514), (9002,4462), (9004,3912), (9026,4358), (9027,14210), (9051,6332), (11477,63).

For Nguyen-Euler centers, X(14226) - X(14245), and a link to Nguyen's research paper, see the preamble just before X(14226).

X(14206) = NGUYEN IMAGE OF X(511)

Barycentrics    cos A - 2 cos B cos C : :
Barycentrics    3 cos A - 2 sin B sin C : :
Barycentrics    b c (-2 a^4+a^2 b^2+b^4+a^2 c^2-2 b^2 c^2+c^4) : :

X(14206) is the intersection of the tangents at X(75) and X(1099) to the inellipse centered at the complement of X(2632). (Randy Hutson, December 2, 2017)

X(14206) lies on these lines: {2,7110}, {19,27}, {100,2688}, {321,3578}, {329,2893}, {514,661}, {896,1109}, {1099,1784}, {1955,4575}, {2166,6149}, {3153,5080}, {3218,4858}, {3219,6358}, {3262,3977}, {3647,6757}, {4647,11684}, {4738,5176}, {6357,7359}

X(14206) = isogonal conjugate of X(2159)
X(14206) = isotomic conjugate of X(2349)
X(14206) = complement of X(18668)
X(14206) = anticomplement of X(18593)
X(14206) = cevapoint of X(30) and X(7359)
X(14206) = X(75)-Ceva conjugate of X(1099)
X(14206) = X(i)-cross conjugate of X(j) for these (i,j): {1099, 75}, {2173, 1784}
X(14206) = trilinear product X(2)*X(30)
X(14206) = trilinear pole of line {1099, 6739, 11125}
X(14206) = polar conjugate of X(36119)
X(14206) = pole wrt polar circle of trilinear polar of X(36119) (line X(19)X(661))
X(14206) = crossdifference of every pair of points on line {31, 810}
X(14206) = crosssum of X(i) and X(j) for these (i,j): {31, 9406}, {41, 3724}, {48, 2315}, {2151, 2152}
X(14206) = X(2986)-aleph conjugate of X(6149)
X(14206) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {80, 2893}, {284, 6224}, {759, 7}, {1793, 4329}, {1807, 2897}, {2161, 2475}, {2341, 8}, {6740, 69}
X(14206) = isoconjugate of X(j) and X(j) for these (i,j): {1, 2159}, {3, 8749}, {6, 74}, {19,35200}, {31, 2349}, {32, 1494}, {50, 5627}, {110, 2433}, {111, 9717}, {186, 11079}, {187, 9139}, {647, 1304}, {1576, 2394}, {3003, 10419}
X(14206) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (896, 1109, 1733), (2580, 2581, 63), (6357, 7359, 11064)
X(14206) = barycentric product X(i)*X(j) for these {i,j}: {1, 3260}, {30, 75}, {69, 1784}, {76, 2173}, {85, 7359}, {92, 11064}, {304, 1990}, {312, 6357}, {561, 1495}, {668, 11125}, {799, 1637}, {811, 9033}, {1099, 1494}, {1502, 9406}, {1577, 2407}, {1928, 9407}, {1969, 3284}, {2166, 6148}, {2631, 6331}
X(14206) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 74}, {2, 2349}, {6, 2159}, {19, 8749}, {30, 1}, {75, 1494}, {113, 1725}, {162, 1304}, {402, 2629}, {661, 2433}, {896, 9717}, {897, 9139}, {1099, 30}, {1109, 12079}, {1495, 31}, {1511, 6149}, {1577, 2394}, {1636, 822}, {1637, 661}, {1650, 2632}, {1749, 3470}, {1784, 4}, {1895, 10152}, {1990, 19}, {2166, 5627}, {2173, 6}, {2407, 662}, {2420, 163}, {2631, 647}, {3163, 2173}, {3260, 75}, {3284, 48}, {4240, 162}, {5642, 896}, {6357, 57}, {6739, 758}, {6793, 2312}, {7359, 9}, {9033, 656}, {9214, 897}, {9406, 32}, {9407, 560}, {9408, 9406}, {9409, 810}, {10272, 1749}, {11064, 63}, {11125, 513}

X(14207) = NGUYEN IMAGE OF X(512)

Barycentrics    b c (b^2-c^2) (-5 a^2+b^2+c^2) : :

X(14207) lies on these lines: {63,798}, {100,2740}, {321,4404}, {514,661}, {5739,6003}

X(14207) = isotomic conjugate of X(37216)
X(14207) = X(8691)-anticomplementary conjugate of X(7)
X(14207) = isoconjugate of X(j) and X(j) for these (i,j): {6, 1296}, {111, 2434}, {1576, 5485}
X(14207) = barycentric product X(i)*X(j) for these {i,j}: {75, 1499}, {321, 4786}, {561, 8644}, {661, 11059}, {799, 6791}, {1577, 1992}
X(14207) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 1296}, {896, 2434}, {1384, 163}, {1499, 1}, {1577, 5485}, {1992, 662}, {2408, 897}, {2444, 923}, {4232, 162}, {4786, 81}, {6791, 661}, {8644, 31}, {9125, 896}, {9126, 6149}, {11059, 799}

X(14208) = NGUYEN IMAGE OF X(520)

Barycentrics    b c (b^2-c^2) (-a^2+b^2+c^2) : :
Barycentrics    (csc A)(sin 2B - sin 2C) : :

X(14208) lies on these lines: {63,822}, {101,2859}, {306,8611}, {514,661}, {656,4025}, {799,4575}, {818,8633}, {850,4077}, {3267,4064}, {4724,4985}

X(14208) = isogonal conjugate of X(32676)
X(14208) = isotomic conjugate of X(162)
X(14208) = X(i)-Ceva conjugate of X(j) for these (i,j): {662, 1930}, {799, 63}, {811, 75}
X(14208) = X(i)-cross conjugate of X(j) for these (i,j): {656, 1577}, {2632, 63}, {4064, 525}
X(14208) = isoconjugate of X(j) and X(j) for these (i,j): {4, 1576}, {6, 112}, {19, 163}, {25, 110}, {28, 692}, {31, 162}, {32, 648}, {51, 933}, {58, 8750}, {99, 1974}, {100, 2203}, {101, 1474}, {107, 184}, {108, 2194}, {109, 2299}, {154, 1301}, {206, 1289}, {232, 2715}, {237, 685}, {249, 2489}, {250, 512}, {427, 4630}, {560, 811}, {577, 6529}, {607, 4565}, {608, 5546}, {643, 1395}, {651, 2204}, {662, 1973}, {667, 5379}, {823, 9247}, {827, 1843}, {906, 5317}, {1096, 4575}, {1172, 1415}, {1297, 2445}, {1304, 1495}, {1333, 1783}, {1414, 2212}, {1461, 2332}, {1501, 6331}, {1625, 8882}, {1897, 2206}, {1976, 4230}, {2189, 4559}, {2207, 4558}, {2211, 2966}, {2333, 4556}, {2393, 10423}, {2420, 8749}, {2576, 2577}, {5467, 8753}, {5994, 8740}, {5995, 8739}, {7115, 7252}, {8541, 11636}
X(14208) = crosspoint of X(i) and X(j) for these (i,j): {75, 811}, {561, 799}
X(14208) = trilinear pole of line {3708, 4466}
X(14208) = crossdifference of every pair of points on line {31, 1932}
X(14208) = crosssum of X(i) and X(j) for these (i,j): {31, 810}, {560, 798}
X(14208) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4077, 4086, 850)
X(14208) = X(345)-beth conjugate of X(8611)
X(14208) = X(163)-zayin conjugate of X(798)
X(14208) = X(8615)-anticomplementary conjugate of X(4440)
X(14208) = trilinear product X(2)*X(525)
X(14208) = trilinear product X(75)*X(656)
X(14208) = trilinear product of Jerabek hyperbola intercepts of de Longchamps line
X(14208) = barycentric product X(i)*X(j) for these {i,j}: {1, 3267}, {63, 850}, {69, 1577}, {72, 3261}, {75, 525}, {76, 656}, {92, 3265}, {125, 799}, {158, 4143}, {274, 4064}, {304, 523}, {305, 661}, {306, 693}, {307, 4391}, {313, 905}, {321, 4025}, {336, 2799}, {338, 4592}, {339, 662}, {345, 4077}, {348, 4086}, {349, 521}, {520, 1969}, {522, 1231}, {561, 647}, {668, 4466}, {670, 3708}, {810, 1502}, {1109, 4563}, {1441, 6332}, {1565, 4033}, {1821, 6333}, {1928, 3049}, {1930, 4580}, {2525, 3112}, {2632, 6331}, {3695, 7199}, {3700, 7182}, {3718, 7178}, {6063, 8611}
X(14208) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 112}, {2, 162}, {3, 163}, {10, 1783}, {37, 8750}, {48, 1576}, {63, 110}, {69, 662}, {71, 692}, {72, 101}, {73, 1415}, {75, 648}, {76, 811}, {77, 4565}, {78, 5546}, {92, 107}, {125, 661}, {158, 6529}, {190, 5379}, {201, 4559}, {226, 108}, {264, 823}, {293, 2715}, {304, 99}, {305, 799}, {306, 100}, {307, 651}, {313, 6335}, {321, 1897}, {326, 4558}, {332, 4612}, {336, 2966}, {337, 4584}, {339, 1577}, {343, 2617}, {345, 643}, {348, 1414}, {394, 4575}, {512, 1973}, {513, 1474}, {514, 28}, {520, 48}, {521, 284}, {522, 1172}, {523, 19}, {525, 1}, {561, 6331}, {647, 31}, {649, 2203}, {650, 2299}, {652, 2194}, {656, 6}, {661, 25}, {662, 250}, {663, 2204}, {684, 1755}, {693, 27}, {798, 1974}, {810, 32}, {822, 184}, {850, 92}, {879, 1910}, {905, 58}, {908, 4246}, {914, 3658}, {1109, 2501}, {1214, 109}, {1231, 664}, {1265, 7259}, {1332, 4570}, {1439, 1461}, {1441, 653}, {1444, 4556}, {1459, 1333}, {1565, 1019}, {1577, 4}, {1650, 2631}, {1799, 4599}, {1812, 4636}, {1821, 685}, {1959, 4230}, {1969, 6528}, {2167, 933}, {2184, 1301}, {2312, 2445}, {2349, 1304}, {2501, 1096}, {2525, 38}, {2533, 7119}, {2574, 2577}, {2575, 2576}, {2582, 1114}, {2583, 1113}, {2592, 2587}, {2593, 2586}, {2616, 8882}, {2618, 53}, {2631, 1495}, {2632, 647}, {2643, 2489}, {2799, 240}, {2968, 1021}, {2972, 822}, {3049, 560}, {3120, 6591}, {3239, 4183}, {3261, 286}, {3265, 63}, {3267, 75}, {3269, 810}, {3657, 913}, {3676, 1396}, {3682, 906}, {3694, 3939}, {3695, 1018}, {3700, 33}, {3708, 512}, {3709, 2212}, {3710, 644}, {3718, 645}, {3737, 2189}, {3900, 2332}, {3912, 4238}, {3926, 4592}, {3936, 4242}, {3942, 3733}, {3949, 4557}, {3998, 1331}, {4010, 2201}, {4017, 608}, {4019, 4579}, {4024, 1824}, {4025, 81}, {4036, 1826}, {4041, 607}, {4064, 37}, {4077, 278}, {4086, 281}, {4088, 5089}, {4091, 1437}, {4129, 4222}, {4131, 1790}, {4143, 326}, {4171, 7071}, {4391, 29}, {4397, 2322}, {4462, 4248}, {4466, 513}, {4468, 4233}, {4551, 7115}, {4552, 7012}, {4558, 1101}, {4560, 270}, {4561, 4567}, {4566, 7128}, {4574, 1110}, {4580, 82}, {4592, 249}, {4705, 2333}, {4707, 1870}, {4841, 5338}, {4988, 2355}, {5489, 3708}, {6332, 21}, {6333, 1959}, {6334, 1725}, {6356, 1020}, {6368, 1953}, {6508, 1624}, {6563, 1748}, {6587, 204}, {6590, 4206}, {7004, 7252}, {7019, 4603}, {7056, 4637}, {7068, 8611}, {7178, 34}, {7180, 1395}, {7182, 4573}, {7216, 1398}, {7253, 2326}, {7254, 849}, {7265, 6198}, {7649, 5317}, {8057, 610}, {8061, 1843}, {8552, 6149}, {8611, 55}, {8673, 2172}, {9033, 2173}, {9409, 9406}, {10097, 923}, {10099, 1438}, {12077, 2181}

X(14209) = NGUYEN IMAGE OF X(525)

Barycentrics    a (a^2-b^2-c^2) (b^2-c^2) (a^4-b^4+4 b^2 c^2-c^4) : :

X(14209) lies on these lines: {63,656}, {514,661}

X(14209) = isotomic conjugate of X(37217)
X(14209) = isoconjugate of X(112) and X(5486)
X(14209 = barycentric product X(656)X(11185)
X(14209) = barycentric quotient X(i)/X(j) for these {i,j}: {656, 5486}, {1995, 162}, {11185, 811}

X(14210) = NGUYEN IMAGE OF X(9027)

Barycentrics    b c (-2 a^2+b^2+c^2) : :

X(14210) lies on these lines: 1,75}, {69,5692}, {85,3760}, {100,2729}, {187,4760}, {190,5525}, {312,3761}, {350,1111}, {514,661}, {519,3263}, {538,4037}, {668,3992}, {712,3726}, {742,3230}, {811,1784}, {896,6629}, {920,1102}, {934,2760}, {1089,1909}, {1926,4602}, {2166,8773}, {2238,8682}, {2241,4372}, {2242,4376}, {3266,4062}, {3685,5088}, {3712,6390}, {3797,7208}, {3930,4568}, {4099,7187}, {4387,7223}, {4459,5148}, {4495,4858}, {4592,6149}, {4645,5195}, {5194,7235}, {6358,7244}, {7176,7283}

X(14210) = midpoint of X(4037) and X(7200)
X(14210) = reflection of X(4986) in X(3263)
X(14210) = isogonal conjugate of X(923)
X(14210) = isotomic conjugate of X(897)
X(14210) = complement of X(17497)
X(14210) = anticomplement of X(16611)
X(14210) = polar conjugate of X(36128)
X(14210) = pole wrt polar circle of trilinear polar of X(36128) (line X(19)X(23894))
X(14210) = X(i)-zayin conjugate of X(j) for these (i,j): {1, 923}, {2642, 798}
X(14210) = X(4062)-cross conjugate of X(524)
X(14210) = isoconjugate of X(j) and X(j) for these (i,j): {1, 923}, {3, 8753}, {6, 111}, {25, 895}, {31, 897}, {32, 671}, {55, 7316}, {56, 5547}, {110, 9178}, {112, 10097}, {187, 10630}, {237, 9154}, {512, 691}, {667, 5380}, {669, 892}, {1296, 2444}, {1495, 9139}, {1576, 5466}, {1976, 5968}, {2393, 10422}, {2715, 8430}, {6137, 9206}, {6138, 9207}
X(14210) = cevapoint of X(524) and X(3712)
X(14210) = trilinear pole of line {2642, 4750}
X(14210) = crossdifference of every pair of points on line {31, 798}
X(14210) = crosssum of X(i) and X(j) for these (i,j): {31, 922}, {3248, 8650}
X(14210) = barycentric product X(i)*X(j) for these {i,j}: {1, 3266}, {75, 524}, {76, 896}, {85, 3712}, {92, 6390}, {187, 561}, {274, 4062}, {304, 468}, {312, 7181}, {321, 6629}, {334, 4760}, {351, 4602}, {668, 4750}, {670, 2642}, {690, 799}, {922, 1502}, {1577, 5468}, {1934, 5026}, {1969, 3292}, {3112, 7813}, {7018, 7267}
X(14210) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 111}, {2, 897}, {6, 923}, {9, 5547}, {19, 8753}, {57, 7316}, {63, 895}, {75, 671}, {187, 31}, {190, 5380}, {351, 798}, {468, 19}, {524, 1}, {656, 10097}, {661, 9178}, {662, 691}, {690, 661}, {799, 892}, {896, 6}, {897, 10630}, {922, 32}, {1577, 5466}, {1648, 2643}, {1649, 2642}, {1821, 9154}, {1959, 5968}, {2349, 9139}, {2482, 896}, {2642, 512}, {3266, 75}, {3292, 48}, {3712, 9}, {4062, 37}, {4235, 162}, {4750, 513}, {4760, 238}, {4831, 1449}, {4933, 45}, {5026, 1580}, {5467, 163}, {5468, 662}, {5477, 8772}, {5642, 2173}, {5967, 1910}, {6390, 63}, {6629, 81}, {7181, 57}, {7267, 171}, {7813, 38}, {9155, 1755}, {9717, 2159}, {11053, 2640}
X(14210) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 304, 1930), (1089, 7278, 1909), (1111, 4975, 350), (3712, 7181, 6390), (4760, 7267, 187)

X(14211) = NGUYEN IMAGE OF X(3)

Barycentrics    b c (-4 a^4+a^2 b^2+3 b^4+a^2 c^2-6 b^2 c^2+3 c^4) : :

X(14211) lies on this line: {19,27}

X(14211) = barycentric product X(75)X(3627)
X(14211) = barycentric quotient X(3627)/X(1)

X(14212) = NGUYEN IMAGE OF X(1350)

Barycentrics    b c (-3 a^4+a^2 b^2+2 b^4+a^2 c^2-4 b^2 c^2+2 c^4) : :

X(14212) lies on these lines: {19,27}, {908,4035}, {1109,1707}

X(14212 = X(5560)-anticomplementary conjugate of X(2893)
X(14212 = isoconjugate of X(6) and X(11270)
X(14212 = barycentric product X(75)X(382)
X(14212 = barycentric quotient X(i)/X(j) for these {i,j}: {1, 11270}, {382, 1}

X(14213) = NGUYEN IMAGE OF X(1351)

Barycentrics    cos(B - C) : :
Barycentrics    b c (-a^2 b^2+b^4-a^2 c^2-2 b^2 c^2+c^4) : :

X(14213) is the intersection of the tangents at X(75) and X(1087) to the inellipse centered at the complement of X(2632). (Randy Hutson, November 2, 2017)

X(14213) lies on the conic {A,B,C,X(91),X(92),X(2621)} and these lines: {1,91}, {2,2006}, {8,2894}, {19,27}, {31,1733}, {38,1109}, {306,3262}, {318,7541}, {321,908}, {662,2167}, {914,1441}, {1087,2181}, {1089,11681}, {1325,2975}, {1821,3112}, {1930,1959}, {1954,4575}, {1969,6521}, {2216,2962}, {3869,4647}, {3874,6757}, {4415,4957}, {4671,5748}, {4692,5176}

X(14213) = isogonal conjugate of X(2148)
X(14213) = isotomic conjugate of X(2167)
X(14213) = trilinear product X(2)*X(5)
X(14213) = polar conjugate X(2190)
X(14213) = X(662)-Ceva conjugate of X(1577)
X(14213) = crosspoint of X(i) and X(j) for these (i,j): {75, 1969}
X(14213) = trilinear pole of line {2618, 6369}
X(14213) = crossdifference of every pair of points on line {810, 8648}
X(14213) = crosssum of X(i) and X(j) for these (i,j): {31, 9247}, {48, 563}
X(14213) = X(75)-waw conjugate of X(1930)
X(14213) = X(i)-zayin conjugate of X(j) for these (i,j): {1, 2148}, {2216, 2180}, {2616, 661}
X(14213) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {79, 2893}, {284, 3648}, {1789, 4329}, {2160, 2475}, {3615, 69}, {7073, 2895}, {7100, 2897}, {7110, 1330}, {8606, 3151}, {13486, 693}
X(14213) = isoconjugate of X(j) and X(j) for these (i,j): {1, 2148}, {3, 8882}, {6, 54}, {19, 2169}, {25, 97}, {31, 2167}, {32, 95}, {47, 2168}, {48, 2190}, {50, 1141}, {96, 571}, {110, 2623}, {163, 2616}, {184, 275}, {186, 11077}, {252, 2965}, {288, 13366}, {570, 1166}, {577, 8884}, {647, 933}, {1298, 1971}
X(14213) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (75, 92, 63), (4858, 6358, 2)
X(14213) = barycentric product X(i)*X(j) for these {i,j}: {1, 311}, {5, 75}, {51, 561}, {53, 304}, {63, 324}, {76, 1953}, {92, 343}, {95, 1087}, {99, 2618}, {216, 1969}, {305, 2181}, {326, 13450}, {799, 12077}, {811, 6368}, {850, 2617}, {1225, 2216}, {1273, 2166}, {1393, 3596}, {1502, 2179}, {6063, 7069}
X(14213) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 54}, {2, 2167}, {3, 2169}, {4, 2190}, {5, 1}, {6, 2148}, {19, 8882}, {51, 31}, {52, 47}, {53, 19}, {63, 97}, {75, 95}, {91, 96}, {92, 275}, {143, 2964}, {158, 8884}, {162, 933}, {216, 48}, {217, 9247}, {311, 75}, {324, 92}, {343, 63}, {467, 1748}, {523, 2616}, {661, 2623}, {920, 8883}, {1087, 5}, {1109, 8901}, {1154, 6149}, {1393, 56}, {1625, 163}, {1749, 1157}, {1953, 6}, {1956, 1298}, {1969, 276}, {2081, 2624}, {2165, 2168}, {2166, 1141}, {2179, 32}, {2180, 571}, {2181, 25}, {2216, 1166}, {2290, 50}, {2313, 1971}, {2595, 2601}, {2596, 2602}, {2599, 2594}, {2600, 654}, {2603, 2597}, {2617, 110}, {2618, 523}, {2962, 252}, {3199, 1973}, {5562, 255}, {6368, 656}, {6369, 3738}, {6521, 8794}, {7069, 55}, {7135, 7134}, {8800, 921}, {12077, 661}, {13157, 2184}, {13450, 158}

X(14214) =  X(30)X(14164)∩X(381)X(14163)

Barycentrics    (a^8+4 a^4 b^4-6 a^2 b^6+b^8-6 a^6 c^2-5 a^4 b^2 c^2+7 a^2 b^4 c^2+10 a^4 c^4+a^2 b^2 c^4-2 b^4 c^4-6 a^2 c^6+c^8) (a^8-6 a^6 b^2+10 a^4 b^4-6 a^2 b^6+b^8-5 a^4 b^2 c^2+a^2 b^4 c^2+4 a^4 c^4+7 a^2 b^2 c^4-2 b^4 c^4-6 a^2 c^6+c^8) : :

X(14214) is a vertex of a rectangle inscribed in the Yff hyperbola; see X(14163).

X(14214) lies on these lines: {30,14164}, {381,14163}

X(14214) = reflection of X(14163) in X(381)
X(14214) = reflection of X(14215) in the Euler line
X(14214) = reflection of X(14164) in line X(381)X(523)

X(14215) =  X(30)X(14163)∩X(381)X(14164)

Barycentrics    (a^8-2 a^4 b^4+b^8-6 a^6 c^2+7 a^4 b^2 c^2+a^2 b^4 c^2-6 b^6 c^2+4 a^4 c^4-5 a^2 b^2 c^4+10 b^4 c^4-6 b^2 c^6+c^8) (a^8-6 a^6 b^2+4 a^4 b^4+b^8+7 a^4 b^2 c^2-5 a^2 b^4 c^2-6 b^6 c^2-2 a^4 c^4+a^2 b^2 c^4+10 b^4 c^4-6 b^2 c^6+c^8) : :

X(14215) is a vertex of a rectangle inscribed in the Yff hyperbola; see X(14163).

X(14215) lies on these lines: {30,14163}, {381,14164}

X(14215) = reflection of X(14164) in X(381)
X(14215) = reflection of X(14214) in the Euler line
X(14215) = reflection of X(14163) in line X(381)X(523)

X(14216) =  X(3)X(66)∩X(4)X(51)

Trilinears    4*cos(B-C)+cos(A)*(2*cos(2*(B- C))-7)+cos(3*A) : :
Barycentrics    a^10-3*(b^2+c^2)*a^8+4*(b^4-b^ 2*c^2+c^4)*a^6-4*(b^4-c^4)*(b^ 2-c^2)*a^4+3*(b^4-c^4)^2*a^2-( b^4-c^4)*(b^2-c^2)^3 : :
X(14216) = 2*X(119)-3*X(1699) = 3*X(165)-4*X(6713) = 2*X(214)-3*X(5603) = 2*X(1145)-3*X(5587) = 4*X(1387)-3*X(3576) = 4*X(1484)-3*X(11219) = 3*X(1699)-X(5541) = 4*X(3035)-5*X(8227) = 3*X(5603)-X(13199) = 3*X(11219)-2*X(12515)

See César Lozada, Hyacinthos 26559

X(14216) lies on these lines: {2, 6759}, {3, 66}, {4, 51}, {5, 1498}, {6, 1595}, {20, 2888}, {30, 64}, {70, 74}, {125, 3542}, {133, 6526}, {140, 154}, {182, 5596}, {184, 3541}, {206, 13336}, {221, 495}, {343, 11414}, {355, 5836}, {381, 2883}, {382, 13093}, {427, 1181}, {485, 12964}, {486, 12970}, {496, 2192}, {499, 10535}, {511, 11411}, {542, 2892}, {548, 8567}, {550, 10606}, {569, 11179}, {578, 3088}, {631, 10282}, {858, 11441}, {1204, 13399}, {1370, 5562}, {1478, 7355}, {1479, 6285}, {1495, 3147}, {1593, 6146}, {1594, 11456}, {1597, 12241}, {1598, 13567}, {1619, 6642}, {1657, 5894}, {1872, 5928}, {1907, 10982}, {2393, 10625}, {2777, 3146}, {2781, 6243}, {2917, 7525}, {3091, 5643}, {3410, 3522}, {3523, 11202}, {3526, 10192}, {3546, 9306}, {3548, 10539}, {3575, 10605}, {3583, 12950}, {3585, 12940}, {3627, 5895}, {3818, 7401}, {3819, 11487}, {3843, 5893}, {4846, 6145}, {5418, 10533}, {5420, 10534}, {5480, 11432}, {5654, 13371}, {5810, 6907}, {5889, 7391}, {5907, 6643}, {6102, 6293}, {6221, 8991}, {6284, 10060}, {6398, 13980}, {6640, 10540}, {6756, 9786}, {7354, 10076}, {7386, 11793}, {7387, 12359}, {7392, 11695}, {7487, 11438}, {7507, 12174}, {7528, 9730}, {7544, 10574}, {8550, 11426}, {10113, 11744}, {10117, 10264}, {10182, 10303}, {10201, 13561}, {10274, 11003}, {10628, 12284}, {11598, 12121}, {11818, 13630}, {12084, 12118}, {12383, 13293}, {12586, 12675}

X(14216) = midpoint of X(i) and X(j) for these {i,j}: {4, 12324}, {382, 13093}, {3146, 12250}, {12317, 13203}
X(14216) = reflection of X(i) in X(j) for these (i,j): (3, 6247), (20, 3357), (1352, 66), (1498, 5), (1657, 5894), (5596, 182), (5878, 4), (5895, 3627), (6193, 13346), (6293, 6102), (7387, 12359), (9833, 3), (9934, 125), (10117, 10264), (11744, 10113), (12118, 12084), (12121, 11598), (12315, 2883), (12383, 13293)
X(14216) = complement of X(34781)
X(14216) = anticomplement of X(6759)
X(14216) = anticomplementary circle-inverse-of-X(6761)
X(14216) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4, 11433, 10110), (4, 11457, 1899), (185, 11550, 4), (381, 12315, 2883), (631, 11206, 10282), (1498, 1853, 5), (1907, 11245, 10982), (3818, 9729, 7401), (7528, 9730, 9815), (11438, 13419, 7487), (12278, 13445, 20)

X(14217) =  REFLECTION OF X(40) IN X(11)

Trilinears    2*sin(A/2)*(6*cos((B-C)/2)-cos (3*(B-C)/2))+3*cos(A)+cos(2*A) -6 : :
Barycentrics    a^7-(2*b^2+3*b*c+2*c^2)*a^5-(b +c)*(b^2-7*b*c+c^2)*a^4+(b^4+c ^4+3*b*c*(b^2-4*b*c+c^2))*a^3+ (b^2-c^2)*(b-c)*(2*b-c)*(b-2* c)*a^2-(b^2-c^2)^3*(b-c) : :
X(14217) = 2*X(119)-3*X(1699) = 3*X(165)-4*X(6713) = 2*X(214)-3*X(5603) = 2*X(1145)-3*X(5587) = 4*X(1387)-3*X(3576) = 4*X(1484)-3*X(11219) = 3*X(1699)-X(5541) = 4*X(3035)-5*X(8227) = 3*X(5603)-X(13199) = 3*X(11219)-2*X(12515)

See César Lozada, Hyacinthos 26559

X(14217) lies on these lines: {1, 5840}, {4, 2802}, {11, 40}, {20, 11715}, {30, 12737}, {46, 5533}, {65, 13274}, {80, 517}, {100, 946}, {104, 516}, {119, 1699}, {149, 151}, {153, 9802}, {165, 6713}, {214, 5603}, {497, 12736}, {515, 1320}, {528, 1537}, {529, 11256}, {952, 3627}, {1145, 5587}, {1317, 1836}, {1387, 3576}, {1482, 7972}, {1484, 5535}, {1768, 9589}, {1770, 10074}, {2077, 10090}, {2093, 12832}, {2099, 12743}, {2809, 10772}, {2817, 10777}, {2829, 6264}, {3035, 8227}, {3057, 13273}, {3149, 13205}, {3585, 12749}, {3898, 6951}, {4295, 5083}, {4301, 10698}, {5057, 12531}, {5119, 8068}, {5180, 12532}, {5443, 11849}, {5657, 6702}, {5660, 12331}, {5697, 10057}, {5805, 9945}, {5854, 5881}, {5856, 11372}, {6284, 11014}, {7743, 13528}, {8148, 12747}, {9612, 10956}, {9616, 13913}, {9897, 11531}, {10058, 11012}, {10087, 12047}, {10265, 10707}, {10624, 10902}, {10993, 11522}, {11571, 12750}, {12608, 13278}, {12619, 12702}, {12672, 13271}, {12763, 13600}

X(14217) = midpoint of X(i) and X(j) for these {i,j}: {149, 962}, {153, 9802}, {1320, 10724}, {1768, 9589}, {5691, 12653}, {8148, 12747}, {9897, 11531}
X(14217) = reflection of X(i) in X(j) for these (i,j): (20, 11715), (40, 11), (80, 10738), (100, 946), (5541, 119), (6326, 1537), (7972, 1482), (10698, 4301), (10993, 11729), (12119, 1), (12331, 12611), (12515, 1484), (12702, 12619), (12751, 4), (13199, 214), (13528, 7743)
X(14217) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4, 11433, 10110), (1484, 12515, 11219), (1699, 5541, 119), (5603, 13199, 214), (9802, 9812, 153), (12331, 12611, 5660), (12700, 12701, 40)

X(14218) =  COMPLEMENT OF X(508)

Barycentrics    a1/2(b1/2sin C/2 + c1/2 sin B/2) : :

See César Lozada, Hyacinthos 26562

X(14218) lies on this line: {2,508}

X(14218) = complement of X(508)

X(14219) =  X(12)X(5620)∩X(101)X(5949)


X(14219) = (2r+R)(rR(2r+5R)-2(2r+R)s^2)) X(12) - (2R(2r^3+7rR(r+R)-(2r+R)s^2) X(5620)

Barycentrics    (b+c)(-a^14 (b+c) - a^13 (b^2+c^2) + a^12 (5 b^3+7 b^2 c+7 b c^2+5 c^3) + 2 a^11 (2b^4+3 b^3 c+6 b^2 c^2+3 b c^3+2 c^4) - a^10 (11 b^5+15 b^4 c+16 b^3 c^2+16 b^2 c^3+15 b c^4+11 c^5) - a^9 (5 b^6+19 b^5 c+35 b^4 c^2+30 b^3 c^3+35 b^2 c^4+19 b c^5+5 c^6) + a^8 (15 b^7+18 b^6 c+2 b^5 c^2-3 b^4 c^3-3 b^3 c^4+2 b^2 c^5+18 b c^6+15 c^7) + a^7 b c (24 b^6+43 b^5 c+26 b^4 c^2+22 b^3 c^3+26 b^2 c^4+43 b c^5+24 c^6) + a^6 (-15 b^9-19 b^8 c+14 b^7 c^2+26 b^6 c^3+13 b^5 c^4+13 b^4 c^5+26 b^3 c^6+14 b^2 c^7-19 b c^8-15 c^9) + a^5 (5 b^10-18 b^9 c-40 b^8 c^2-5 b^7 c^3+12 b^6 c^4+2 b^5 c^5+12 b^4 c^6-5 b^3 c^7-40 b^2 c^8-18 b c^9+5 c^10) + a^4 (11 b^11+15 b^10 c-19 b^9 c^2-38 b^8 c^3-10 b^7 c^4+21 b^6 c^5+21 b^5 c^6-10 b^4 c^7-38 b^3 c^8-19 b^2 c^9+15 b c^10+11 c^11) - a^3 (b^2-c^2)^2 (4 b^8-10 b^7 c-21 b^6 c^2-7 b^5 c^3+5 b^4 c^4-7 b^3 c^5-21 b^2 c^6-10 b c^7+4 c^8) - a^2 (b-c)^2 (b+c)^3 (5 b^8-8 b^6 c^2-7 b^5 c^3+16 b^4 c^4-7 b^3 c^5-8 b^2 c^6+5 c^8) + a (b^2-c^2)^4 (b^6-3 b^5 c-4 b^4 c^2+4 b^3 c^3-4 b^2 c^4-3 b c^5+c^6) + (b-c)^6 (b+c)^7 (b^2-b c+c^2)) : :

See Tsihong Lau and Angel Montesdeoca, Quadri-Figures 2589, and Angel Montesdeoca, Image and details

X(14219) lies on these lines: {12, 5620}, {101, 5949}, {442, 5953}

X(14220) =  ISOGONAL CONJUGATE OF X(7480)

Barycentrics    (b^2-c^2) (-a^2+b^2+c^2) (a^8+a^6 b^2-4 a^4 b^4+a^2 b^6+b^8-3 a^6 c^2+2 a^4 b^2 c^2+2 a^2 b^4 c^2-3 b^6 c^2+3 a^4 c^4-2 a^2 b^2 c^4+3 b^4 c^4-a^2 c^6-b^2 c^6) (-a^8+3 a^6 b^2-3 a^4 b^4+a^2 b^6-a^6 c^2-2 a^4 b^2 c^2+2 a^2 b^4 c^2+b^6 c^2+4 a^4 c^4-2 a^2 b^2 c^4-3 b^4 c^4-a^2 c^6+3 b^2 c^6-c^8) : :

See Bernard Gibert, Cubics: K186.

Let A', B', C' be points on the circumcircle such that AA' || BB' || CC' || Euler line. (i.e., A'B'C' is the circumcevian triangle of X(30).) Let A0B0C0 be the triangle bounded by the Simson lines of A', B', C'. The triangle A0B0C0 is perspective to ABC at X(14220). (See ADGEOM #1674, Dao Thanh Oai, 9/11/2014 and related messages). (Randy Hutson, November 2, 2017)

X(14220) lies on the Jerabek hyperbola, the cubic X186, and these lines: 3,9033}, {4,526}, {6,1637}, {67,8675}, {74,477}, {265,520}, {512,10293}, {690,3426}, {895,9007}, {924,11744}, {2411,3431}, {4846,9517}, {5504,8057}, {6368,11559}

X(14220) = isogonal conjugate of X(7480)
X(14220) = X(i)-isoconjugate of X(j) for these (i,j): {1, 7480}, {162, 5663}
X(14220) = barycentric product X(i)*X(j) for these {i,j}: {265, 2411}, {328, 2436}, {477, 525}
X(14220) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 7480}, {265, 2410}, {477, 648}, {647, 5663}, {1637, 11251}, {2411, 340}, {2436, 186}
X(14220) = orthocenter of X(3)X(4)X(74)
X(14220) = antigonal image of isogonal conjugate of X(4240)

X(14221) =  REFLECTION OF X(316) IN X(3260)

Barycentrics    (a^2-b^2) (a^2-c^2) (a^6 b^2-a^4 b^4-a^2 b^6+b^8+a^6 c^2-2 a^4 b^2 c^2+2 a^2 b^4 c^2-3 b^6 c^2-a^4 c^4+2 a^2 b^2 c^4+4 b^4 c^4-a^2 c^6-3 b^2 c^6+c^8) : :

See Bernard Gibert, Cubics: K186.

X(14221) lies on the cubic K186 and these lines: {4,69}, {99,523}, {107,10425}, {249,648}, {850,4576}

Let P=X(110), the focus of Kiepert parabola, and ℓ1. ℓ2 perpendicular lines through P, that intersect the sidelines BC, CA, AB at points A1 and A2, B1 and B2, C1 and C2, respectively Let Abc = A1C2 ∩ A2B1 and Acb = A1B2 ∩ A2C1. When the lines ℓ1 and ℓ2 rotate around P, the points Abc and Acb lie on the same conic C(A), through B and C. Let Ao be its center. The conics C(B), C(C) and and their centers Bo and Co are defined cyclically. Then the lines AAo , BBo and CCo concur in X(14221). (See HG060721). (Angel Montesdeoca, July 13, 2021)

X(14221) = isotomic conjugate of the isogonal conjugate of X(7468)
X(14221) = reflection of X(316) in X(3260)
X(14221) = crosspoint of X(670) and X(6035)
X(14221) = crosssum of X(669) and X(6041)
X(14221) = barycentric product X(i)*X(j) for these {i,j}: {76,7468}, {670,2493}
X(14221) = barycentric quotient X(i)/X(j) for these {i,j}: {2493,512}, {7468,6}

X(14222) =  X(4)X(924)∩X(107)X(250)

Barycentrics    b^2-c^2) (a^2+b^2-c^2)^2 (a^2-b^2-b c-c^2) (a^2-b^2+b c-c^2) (a^2-b^2+c^2)^2 (a^6-a^4 b^2-a^2 b^4+b^6-2 a^4 c^2+2 a^2 b^2 c^2-2 b^4 c^2+a^2 c^4+b^2 c^4) (a^6-2 a^4 b^2+a^2 b^4-a^4 c^2+2 a^2 b^2 c^2+b^4 c^2-a^2 c^4-2 b^2 c^4+c^6) : :

See Bernard Gibert, Cubics: K186.

X(14222) lies on the cubic K186 and these lines: {4,924}, {107,250}, {523,1300}, {526,5962}

X(14223) =  X(2)X(1637)∩X(4)X(690)

Barycentrics    (b^2-c^2) (a^6-a^4 b^2-a^2 b^4+b^6-a^4 c^2-b^4 c^2+2 a^2 c^4+2 b^2 c^4-2 c^6) (-a^6+a^4 b^2-2 a^2 b^4+2 b^6+a^4 c^2-2 b^4 c^2+a^2 c^4+b^2 c^4-c^6) : :

See Bernard Gibert, Cubics: K186.

X(14223) lies on the Kiepert hyperbola, the cubic K186, and these lines: {2,1637}, {4,690}, et al}

X(14223) = reflection of X(2394) in X(115)
X(14223) = isotomic conjugate of X(14999)
X(14223) = X(6035)-Ceva conjugate of X(5641)
X(14223) = X(1640)-cross conjugate of X(523)
X(14223) = polar conjugate of X(7473)
X(14223) = cevapoint of X(523) and X(1640)
X(14223) = crosspoint of X(5641) and X(6035)
X(14223) = trilinear pole of line {523, 868}
X(14223) = crosssum of X(5191) and X(6041)
X(14223) = antigonal image of X(2394)
X(14223) = syngonal conjugate of X(5664)
X(14223) = Kiepert hyperbola antipode of X(2394)
X(14223) = pole wrt polar circle of trilinear polar of X(7473) (line X(542)X(6103))
X(14223) = orthocenter of X(2)X(4)X(98)
X(14223) = orthocenter of X(4)X(13)X(14)
X(14223) = X(3737)-zayin conjugate of X(2247)
X(14223) = X(i)-isoconjugate of X(j) for these (i,j): {48, 7473}, {110, 2247}, {163, 542}, {662, 5191}, {1101, 1640}, {4575, 6103}
X(14223) = barycentric product X(i)*X(j) for these {i,j}: {115, 6035}, {338, 5649}, {523, 5641}, {842, 850}
X(14223) = barycentric quotient X(i)/X(j) for these {i,j}: {4, 7473}, {115, 1640}, {512, 5191}, {523, 542}, {661, 2247}, {842, 110}, {2501, 6103}, {3124, 6041}, {5641, 99}, {5649, 249}, {6035, 4590}

X(14224) =  X(1)X(11125)∩X(104)X(523)

Barycentrics    (b-c) (-a+b+c) (a^6-a^4 b^2-a^2 b^4+b^6-a^5 c+2 a^4 b c-a^3 b^2 c-a^2 b^3 c+2 a b^4 c-b^5 c-2 a^4 c^2+2 a^2 b^2 c^2-2 b^4 c^2+2 a^3 c^3-a^2 b c^3-a b^2 c^3+2 b^3 c^3+a^2 c^4+b^2 c^4-a c^5-b c^5) (-a^6+a^5 b+2 a^4 b^2-2 a^3 b^3-a^2 b^4+a b^5-2 a^4 b c+a^2 b^3 c+b^5 c+a^4 c^2+a^3 b c^2-2 a^2 b^2 c^2+a b^3 c^2-b^4 c^2+a^2 b c^3-2 b^3 c^3+a^2 c^4-2 a b c^4+2 b^2 c^4+b c^5-c^6) : :

See Bernard Gibert, Cubics: K186.

X(14224) lies on the Feuerbach hyperbola, the cubic K186, and these lines: {1,11125}, {4,8674}, {21,2804}, {79,3738}, {104,523}, {521,11604}, {522,3065}, {900,10308}, {6596,8058}

X(14224) = X(7253)-beth conjugate of X(3065)
X(14224) = X(109)-isoconjugate of X(2771)
X(14224) = barycentric product X(2687)X(4391)
X(14224) = barycentric quotient X(i)/X(j) for these {i,j}: {650, 2771}, {2687, 651}

X(14225) =  X(4)X(14050)∩X(523)X(1141)

Barycentrics    (b^2-c^2) (-a^2 b^2+b^4-a^2 c^2-2 b^2 c^2+c^4) (a^10-3 a^8 b^2+2 a^6 b^4+2 a^4 b^6-3 a^2 b^8+b^10-2 a^8 c^2+3 a^6 b^2 c^2-2 a^4 b^4 c^2+3 a^2 b^6 c^2-2 b^8 c^2+2 a^6 c^4+a^4 b^2 c^4+a^2 b^4 c^4+2 b^6 c^4-4 a^4 c^6-6 a^2 b^2 c^6-4 b^4 c^6+5 a^2 c^8+5 b^2 c^8-2 c^10) (-a^10+2 a^8 b^2-2 a^6 b^4+4 a^4 b^6-5 a^2 b^8+2 b^10+3 a^8 c^2-3 a^6 b^2 c^2-a^4 b^4 c^2+6 a^2 b^6 c^2-5 b^8 c^2-2 a^6 c^4+2 a^4 b^2 c^4-a^2 b^4 c^4+4 b^6 c^4-2 a^4 c^6-3 a^2 b^2 c^6-2 b^4 c^6+3 a^2 c^8+2 b^2 c^8-c^10) : :

See Bernard Gibert, Cubics: K186.

X(14225) lies on the conics {A,B,C,X(4),X(5)}, the cubic K186, and these lines: {4,14050}, {523,1141}, {1263,6368}

leftri

Nguyen-Euler centers: X(14226)-X(14245)

rightri

This preamble and centers X(14226)-X(14245) were contributed by César Eliud Lozada, September 4, 2017.

This section is based on Nguyen Ngoc Giang's paper mentioned in the preamble just before X(14206). The paper appears in the International Journal of Computer Discovered Mathematics, vol 2(2017), pp 135-140.

Let BBACAC, CCBABA, AACBCB be three rectangles built on the sides of a triangle ABC, such that |BBA|/a = |CCB|/b = |AAC|/c = λ = constant. Assume that λ %gt; 0 means that rectangles are built outwards from ABC and λ < 0 means that rectangles are built inwards from ABC. Nguyen proved that if NA, NB, NC are X(5) (i.e., the nine-point-center) of triangles ABACA, BCBAB and CACBC, respectively, then triangles NANBNC and ABC are orthologic.

In addition, it can be proved that if the three nine-point-centers (the N's) are replaced by any other point P on the Euler line such that (X(3)P)|/(X(3)X(4)) = (OP)/(OH) = t, a constant invariant of (a,b,c), then the triangles PAPBPC and ABC remain orthologic. In this case, the orthologic center PAPBPC to ABC is:

  X' = (2*a^6 - (b^2+c^2)*(a^4+(b^2-c^2)^2))*(3*t-1)*λ - 2*S*(a^2+b^2-c^2)*(a^2-b^2+c^2) : : (barycentrics)

and the orthologic center ABC to PAPBPC:

  X" = 1/(((2*a^4 + 2*(b^2+c^2)*a^2 + 4*b^2*c^2 - 4*(b^2+c^2)^2)*t + (a^2+b^2+c^2)*(-a^2+b^2+c^2))*λ + 2*S*(-a^2+b^2+c^2)) : : (barycentrics)

The point X' is here named the Nguyen-Euler(λ) point of P, and the point X", the Nguyen-Euler(λ) adjoint point of P.

For every λ and P,:

X' lies on the van Aubel line X(4)X(6), and |X(4)X'|/|X(4)X(6)| = λ*(3*t - 1)*cot(ω), where ω is the Brocard angle of ABC; moreover, X" lies on the Kiepert hyperbola.

The appearance of (n, I, J) in the following lists means that, for the given λ, the Nguyen-Euler(λ) point and Nguyen-Euler(λ) adjoint point of X(n) are X(I) and X(J), respectively:

The appearance of (I, t) in the following list (complete up to I≤14213) means that X(I), on the Euler line, satisfies |OX(I)|/|OH| = t :
(2, 1/3), (3, 0), (4, 1), (5, 1/2), (20, -1), (140, 1/4), (376, -1/3), (381, 2/3), (382, 2), (546, 3/4), (547, 5/12), (548, -1/4), (549, 1/6), (550, -1/2), (631, 1/5), (632, 3/10), (1656, 2/5), (1657, -2), (2041, 2+sqrt(3)), (2042, 2-sqrt(3)), (2043, -sqrt(3)/3), (2044, sqrt(3)/3), (2045, (4-sqrt(3))/13), (2046, (4+sqrt(3))/13), (2675, (-15+24*sqrt(5))/59), (2676, (75-12*sqrt(5))/109), (3090, 3/7), (3091, 3/5), (3146, 3), (3522, -1/5), (3523, 1/7), (3524, 1/9), (3525, 3/11), (3526, 2/7), (3528, -1/7), (3529, -3), (3530, 1/8), (3533, 5/17), (3534, -2/3), (3543, 5/3), (3544, 9/17), (3545, 5/9), (3627, 3/2), (3628, 3/8), (3830, 4/3), (3832, 5/7), (3839, 7/9), (3843, 4/5), (3845, 5/6), (3850, 5/8), (3851, 4/7), (3853, 5/4), (3854, 11/17), (3855, 7/11), (3856, 11/16), (3857, 9/14), (3858, 7/10), (3859, 13/20), (3860, 17/24), (3861, 7/8), (5054, 2/9), (5055, 4/9), (5056, 5/11), (5059, -5), (5066, 7/12), (5067, 5/13), (5068, 7/13), (5070, 4/11), (5071, 7/15), (5072, 6/11), (5073, 4), (5076, 6/5), (5079, 6/13), (7486, 7/17), (8703, -1/6), (10109, 11/24), (10124, 7/24), (10299, 1/13), (10303, 3/13), (10304, -1/9), (11001, -5/3), (11539, 5/18), (11540, 13/48), (11541, 9), (11737, 13/24), (11812, 5/24), (12100, 1/12), (12101, 13/12), (12102, 9/8), (12103, -3/4), (12108, 3/16), (12811, 9/16), (12812, 9/20), (14093, -2/15)

X(14226) = NGUYEN-EULER(-2) ADJOINT POINT OF X(2)

Barycentrics    (2*S-3*SB)*(2*S-3*SC) : :
X(14226) = 13*X(1132)+2*X(6408) = 2*X(1132)+X(13939) = 7*X(1132)+2*X(13961) = 4*X(6408)-13*X(13939) = 7*X(6408)-13*X(13961) = 7*X(13939)-4*X(13961)

X(14226) lies on the Kiepert hyperbola and these lines: {2,6221}, {3,3591}, {4,3594}, {5,3590}, {6,14241}, {30,1132}, {226,1371}, {372,12819}, {376,486}, {381,1131}, {485,3545}, {542,6569}, {631,10194}, {632,9693}, {1327,6436}, {1328,3069}, {1588,3316}, {3071,3317}, {3090,10195}, {3525,10147}, {3533,6482}, {3534,13941}, {3839,7584}, {3845,7586}, {5870,14243}, {6395,12101}, {6417,11737}, {6434,11001}, {6445,11540}, {7376,10159}, {8972,10109}, {9862,13968}, {10304,13951}, {10783,14238}, {10784,13932}, {13678,13757}, {13748,14228}, {13794,14227}

X(14226) = isogonal conjugate of X(6398)

X(14227) = NGUYEN-EULER(-2) POINT OF X(4)

Barycentrics    S*(a^2+b^2-c^2)*(a^2-b^2+c^2)+4*a^6-2*(b^2+c^2)*a^4-2*(b^4-c^4)*(b^2-c^2) : :
X(14227) = 3*X(4)-2*X(5871) = 3*X(4)-4*X(13748) = 5*X(4)-4*X(13749) = 9*X(4)-8*X(14230) = 7*X(4)-8*X(14233) = 17*X(4)-16*X(14235) = 15*X(4)-16*X(14239) = 3*X(13794)-2*X(14237)

X(14227) lies on these lines: {2,14228}, {4,6}, {376,13712}, {486,14243}, {3316,14232}, {3317,3424}, {3529,12510}, {3535,11206}, {7000,13939}, {13794,14226}

X(14227) = reflection of X(i) in X(j) for these (i,j): (4, 5870), (5871, 13748), (14242, 4)
X(14227) = X(14227) = Nguyen-Euler(-2) point of X(20)
X(14227) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4, 6776, 7581), (4, 10784, 7582), (5870, 5871, 13748), (5871, 13748, 4), (13748, 13749, 14239)

X(14228) = NGUYEN-EULER(-2) ADJOINT POINT OF X(4)

Barycentrics    (SB*(S+4*SW)-2*S^2)*(SC*(S+4*SW)-2*S^2) : :

X(14228) lies on the Kiepert hyperbola and these lines: {2,14227}, {262,10784}, {485,14242}, {1503,3316}, {3317,5870}, {5871,14241}, {13748,14226}

X(14228) = Nguyen-Euler(-2) adjoint point of X(20)

X(14229) = NGUYEN-EULER(-2) ADJOINT POINT OF X(5)

Barycentrics    (SB*(S+SW)-S^2)*(SC*(S+SW)-S^2) : :
X(14229) = 2*X(6222)+3*X(13794) = X(12257)+3*X(13794)

X(14229) lies on the Kiepert hyperbola and these lines: {2,6222}, {3,5490}, {4,6423}, {76,488}, {98,5870}, {262,1588}, {275,3127}, {485,6776}, {486,13880}, {487,8781}, {671,8982}, {1327,6250}, {1503,13881}, {1587,14245}, {2052,5200}, {2996,12256}, {3316,10784}, {3564,5491}, {5485,12510}, {5871,14238}, {6118,10515}, {9757,10577}, {10783,13850}, {13834,14237}

X(14229) = reflection of X(12257) in X(6222)
X(14229) = isogonal conjugate of X(9733)

X(14230) = NGUYEN-EULER(-2) POINT OF X(140)

Barycentrics    4*S*(a^2+b^2-c^2)*(a^2-b^2+c^2)-2*a^6+(b^2+c^2)*a^4+(b^4-c^4)*(b^2-c^2) : :
X(14230) = 5*X(4)-X(5870) = 3*X(4)+X(5871) = 3*X(4)-X(13748) = 9*X(4)-X(14227) = 3*X(4)-2*X(14239)

X(14230) lies on these lines: {4,6}, {5,6119}, {30,9738}, {382,12313}, {485,13644}, {490,7773}, {615,8982}, {1327,14237}, {1350,12323}, {1586,10192}, {6239,9973}, {6251,7584}, {6813,10837}, {7374,9756}, {11477,12222}, {12818,14232}

X(14230) = midpoint of X(i) and X(j) for these {i,j}: {4, 13749}, {5871, 13748}
X(14230) = reflection of X(i) in X(j) for these (i,j): (4, 14235), (13748, 14239), (14233, 4)
X(14230) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4, 3070, 5480), (4, 5871, 13748), (4, 13748, 14239), (13748, 13749, 5871), (13748, 14239, 14233), (13749, 14235, 14233)

X(14231) = NGUYEN-EULER(-2) ADJOINT POINT OF X(140)

Barycentrics    (SB*(2*S-SW)-S^2)*(SC*(2*S-SW)-S^2) : :

X(14231) lies on the Kiepert hyperbola and these lines: {2,9738}, {4,1505}, {6,14238}, {98,3071}, {637,5491}, {1588,14244}, {5111,5480}, {9873,14232}

X(14232) = NGUYEN-EULER(-1) ADJOINT POINT OF X(4)

Barycentrics    (SB*(S+2*SW)-S^2)*(SC*(S+2*SW)-S^2) : :
X(14232) = 3*X(1327)-2*X(13749) = 3*X(13674)-X(14242)

X(14232) lies on the Kiepert hyperbola and these lines: {2,5870}, {76,490}, {485,1503}, {486,6399}, {1131,5871}, {1327,13749}, {1328,14233}, {2996,12297}, {3316,14227}, {3767,14237}, {5490,12322}, {5491,6281}, {9873,14231}, {12818,14230}, {13674,14241}, {13711,14244}

X(14232) = isogonal conjugate of X(11825)

X(14233) = NGUYEN-EULER(-1) POINT OF X(5)

Barycentrics    4*S*(a^2+b^2-c^2)*(a^2-b^2+c^2)+2*a^6-(b^2+c^2)*a^4-(b^4-c^4)*(b^2-c^2) : :
X(14233) = 3*X(4)+X(5870) = 5*X(4)-X(5871) = 3*X(4)-X(13749) = 7*X(4)+X(14227) = 3*X(4)-2*X(14235)

X(14233) lies on these lines: {4,6}, {5,6118}, {30,9739}, {382,12314}, {486,13763}, {489,7773}, {1328,14232}, {1350,12322}, {1585,10192}, {6250,7583}, {6400,9973}, {6811,10838}, {7000,9756}, {11477,12221}, {12819,14237}

X(14233) = midpoint of X(i) and X(j) for these {i,j}: {4, 13748}, {5870, 13749}
X(14233) = reflection of X(i) in X(j) for these (i,j): (4, 14239), (13749, 14235), (14230, 4)
X(14233) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4, 3071, 5480), (4, 5870, 13749), (4, 13749, 14235), (13748, 13749, 5870), (13748, 14239, 14230), (13749, 14235, 14230)

X(14234) = NGUYEN-EULER(-1) ADJOINT POINT OF X(5)

Barycentrics    (SB*(2*S+SW)-S^2)*(SC*(2*S+SW)-S^2) : :

X(14234) lies on the Kiepert hyperbola and these lines: {4,5062}, {6,14245}, {98,13748}, {262,3071}, {486,2459}, {1131,6776}, {1503,14238}, {3070,14240}, {5870,14244}, {10784,14241}

X(14234) = isogonal conjugate of X(9739)

X(14235) = NGUYEN-EULER(-1) POINT OF X(140)

Barycentrics    -8*S*(a^2+b^2-c^2)*(a^2-b^2+c^2)+2*a^6-(b^2+c^2)*a^4-(b^4-c^4)*(b^2-c^2) : :
X(14235) = 9*X(4)-X(5870) = 7*X(4)+X(5871) = 5*X(4)-X(13748) = 3*X(4)+X(13749) = 17*X(4)-X(14227) = 3*X(4)-X(14233)

X(14235) lies on these lines: {4,6}, {5,12975}, {3629,12601}, {6239,12061}

X(14235) = midpoint of X(i) and X(j) for these {i,j}: {4, 14230}, {13749, 14233}
X(14235) = reflection of X(14239) in X(4)
X(14235) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4, 13749, 14233), (13748, 13749, 14242), (14230, 14233, 13749)

X(14236) = NGUYEN-EULER(-1) ADJOINT POINT OF X(140)

Barycentrics    (SB*(4*S-SW)-S^2)*(SC*(4*S-SW)-S^2) : :

X(14236) lies on the Kiepert hyperbola and these lines: {1132,8982}, {3071,14238}, {5480,14240}

X(14236) = isogonal conjugate of X(12975)

X(14237) = NGUYEN-EULER(1) ADJOINT POINT OF X(4)

Barycentrics    (SB*(S-2*SW)+S^2)*(SC*(S-2*SW)+S^2) : :
X(14237) = 3*X(1328)-2*X(13748) = 3*X(13794)-X(14227)

X(14237) lies on the Kiepert hyperbola and these lines: {2,5871}, {76,489}, {485,6222}, {486,1503}, {1132,5870}, {1327,14230}, {1328,13748}, {2996,12296}, {3317,14242}, {3767,14232}, {5490,6278}, {5491,12323}, {9873,14245}, {12819,14233}, {13794,14226}, {13834,14229}

X(14237) = isogonal conjugate of X(11824)

X(14238) = NGUYEN-EULER(1) ADJOINT POINT OF X(5)

Barycentrics    (SB*(2*S-SW)+S^2)*(SC*(2*S-SW)+S^2) : :

X(14238) lies on the Kiepert hyperbola and these lines: {2,8982}, {4,5058}, {6,14231}, {98,13749}, {262,3070}, {485,2460}, {1132,6776}, {1503,14234}, {3071,14236}, {5871,14229}, {10783,14226}

X(14238) = isogonal conjugate of X(9738)

X(14239) = NGUYEN-EULER(1) POINT OF X(140)

Barycentrics    8*S*(a^2+b^2-c^2)*(a^2-b^2+c^2)+2*a^6-(b^2+c^2)*a^4-(b^4-c^4)*(b^2-c^2) : :
X(14239) = 7*X(4)+X(5870) = 9*X(4)-X(5871) = 3*X(4)+X(13748) = 5*X(4)-X(13749) = 15*X(4)+X(14227) = 3*X(4)-X(14230)

X(14239) lies on these lines: {4,6}, {5,12974}, {3629,12602}, {6400,12061}

X(14239) = midpoint of X(i) and X(j) for these {i,j}: {4, 14233}, {13748, 14230}
X(14239) = reflection of X(14235) in X(4)
X(14239) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4, 13748, 14230), (13748, 13749, 14227), (14230, 14233, 13748)

X(14240) = NGUYEN-EULER(1) ADJOINT POINT OF X(140)

Barycentrics    (SB*(4*S+SW)+S^2)*(SC*(4*S+SW)+S^2) : :

X(14240) lies on the Kiepert hyperbola and these lines: {3070,14234}, {5480,14236}

X(14240) = isogonal conjugate of X(12974)

X(14241) = NGUYEN-EULER(2) ADJOINT POINT OF X(2)

Barycentrics    (2*S+3*SB)*(2*S+3*SC) : :
X(14241) = 13*X(1131)+2*X(6407) = 2*X(1131)+X(13886) = 7*X(1131)+2*X(13903) = 4*X(6407)-13*X(13886) = 7*X(6407)-13*X(13903) = 7*X(13886)-4*X(13903)

X(14241) lies on the Kiepert hyperbola and these lines: {2,6398}, {3,3590}, {4,3592}, {5,3591}, {6,14226}, {30,1131}, {226,1372}, {371,12818}, {376,485}, {381,1132}, {486,3545}, {542,6568}, {631,10195}, {1327,3068}, {1328,6435}, {1587,3317}, {1991,5485}, {3070,3316}, {3090,10194}, {3525,10148}, {3533,6483}, {3534,8972}, {3839,7583}, {3845,7585}, {5871,14228}, {6199,12101}, {6418,11737}, {6433,11001}, {6446,11540}, {6490,9541}, {7375,10159}, {8976,10304}, {9862,13908}, {10109,13941}, {10783,13850}, {10784,14234}, {13637,13798}, {13674,14232}

X(14241) = isogonal conjugate of X(6221)

X(14242) = NGUYEN-EULER(2) POINT OF X(4)

Barycentrics    -S*(a^2+b^2-c^2)*(a^2-b^2+c^2)+4*a^6-2*(b^2+c^2)*a^4-2*(b^4-c^4)*(b^2-c^2) : :
X(14242) = 3*X(4)-2*X(5870) = 5*X(4)-4*X(13748) = 3*X(4)-4*X(13749) = 7*X(4)-8*X(14230) = 9*X(4)-8*X(14233) = 15*X(4)-16*X(14235) = 3*X(13674)-2*X(14232)

X(14242) lies on these lines: {2,14243}, {4,6}, {376,13835}, {485,14228}, {3316,3424}, {3317,14237}, {3529,12509}, {3536,11206}, {7374,13886}, {13674,14232}

X(14242) = reflection of X(i) in X(j) for these (i,j): (4, 5871), (5870, 13749), (14227, 4)
X(14242) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4, 6776, 7582), (4, 10783, 7581), (5870, 5871, 13749), (5870, 13749, 4), (13748, 13749, 14235)

X(14243) = NGUYEN-EULER(2) ADJOINT POINT OF X(4)

Barycentrics    (SB*(S-4*SW)+2*S^2)*(SC*(S-4*SW)+2*S^2) : :

X(14243) lies on the Kiepert hyperbola and these lines: {2,14242}, {262,10783}, {486,14227}, {1503,3317}, {3316,5871}, {5870,14226}, {13749,14241}

X(14244) = NGUYEN-EULER(2) ADJOINT POINT OF X(5)

Barycentrics    (SB*(S-SW)+S^2)*(SC*(S-SW)+S^2) : :
X(14244) = 2*X(6399)+3*X(13674) = X(12256)+3*X(13674)

X(14244) lies on the Kiepert hyperbola and these lines: {2,6290}, {3,5491}, {4,6424}, {76,487}, {98,5871}, {262,1587}, {275,3128}, {485,13921}, {486,6776}, {488,8781}, {671,12296}, {1328,6251}, {1503,13881}, {1588,14231}, {2996,12257}, {3317,10783}, {3564,5490}, {5485,12509}, {5870,14234}, {6119,10514}, {9758,10576}, {10784,13932}, {13711,14232}

X(14244) = reflection of X(12256) in X(6399)
X(14244) = isogonal conjugate of X(9732)

X(14245) = NGUYEN-EULER(2) ADJOINT POINT OF X(140)

Barycentrics    (SB*(2*S+SW)+S^2)*(SC*(2*S+SW)+S^2) : :

X(14245) lies on the Kiepert hyperbola and these lines: {2,9739}, {4,1504}, {6,14234}, {98,3070}, {638,5490}, {1587,14229}, {5111,5480}, {9873,14237}

X(14246) = X(3)X(691)∩X(4)X(542)

Barycentrics    a^2*(a^2 + b^2 - 2*c^2)*(a^2 - 2*b^2 + c^2)*(a^4 - b^4 + b^2*c^2 - c^4) : :

X(14246) lies on the cubics K028, K072, K283, K582, and these lines: {2, 8877}, {3, 691}, {4, 542}, {6, 10630}, {8, 5380}, {32, 111}, {76, 892}, {83, 5466}, {315, 6328}, {316, 10510}, {1995, 10422}, {2896, 8561}, {3292, 10559}, {5169, 10415}, {9178, 11638}, {9716, 10560}, {10558, 11422}

X(14246) = X(892)-Ceva conjugate of X(9979)
X(14246) = X(i)-cross conjugate of X(j) for these (i,j): {6593, 23}, {9517, 691}
X(14246) = crosssum of X(2482) and X(7813)
X(14246) = cevapoint of X(23) and X(6593)
X(14246) = X(i)-isoconjugate of X(j) for these (i,j): {67, 896}, {524, 2157}, {3455, 14210}
X(14246) = trilinear pole of line {23, 2492}
X(14246) = isogonal conjugate of X(14357)
X(14246) = Cundy-Parry Phi transform of X(842)
X(14246) = Cundy-Parry Psi transform of X(542)
X(14246) = trilinear product of vertices of 2nd Parry triangle
X(14246) = barycentric product X(i)*X(j) for these {i,j}: {23, 671}, {99, 10561}, {111, 316}, {249, 10555}, {691, 9979}, {892, 2492}, {7664, 10630}
X(14246) = barycentric quotient X(i)/X(j) for these {i,j}: {23, 524}, {111, 67}, {316, 3266}, {923, 2157}, {2492, 690}, {6593, 2482}, {8744, 468}, {8753, 8791}, {9019, 7813}, {10317, 3292}, {10555, 338}, {10561, 523}, {10630, 10415}

X(14247) = X(3)X(827)∩X(4)X(83)

Barycentrics    a^2*(a^2 + b^2)*(a^2 + c^2)*(a^4 - b^4 - b^2*c^2 - c^4) : :

X(14247) lies on the cubic K028 and these lines: {3, 827}, {4, 83}, {76, 4577}, {184, 7877}, {206, 3096}, {574, 9481}, {3730, 4628}, {7502, 9821}

X(14247) = isogonal conjugate of X(14378)
X(14247) = X(1930)-isoconjugate of X(3456)
X(14247) = barycentric product X(i)*X(j) for these {i,j}: {83, 6636}, {251, 7768}
X(14247) = barycentric quotient X(i)/X(j) for these {i,j}: {6636, 141}, {7768, 8024}

X(14248) = X(3)X(2971)∩X(4)X(193)

Barycentrics    a^2(tan A)/(cot B + cot C - cot A) : :
Barycentrics    a^2*(a^2 + b^2 - 3*c^2)*(a^2 + b^2 - c^2)*(a^2 - 3*b^2 + c^2)*(a^2 - b^2 + c^2) : :

The following four circles intersect in two points, and their cevapoint is X(14248): (1) circumcircle, (2) 2nd Lemoine circle, (3) {X(371),X(372),PU(1),PU(39)} (whose center is X(32)), and (4) {X(4),X(194),X(3557),X(3558)} (whose center is X(3095). (Randy Hutson, November 2, 2017)

X(14248) lies on the cubics K028, K164, K233, K708 and these lines: {3, 2971}, {4, 193}, {25, 1611}, {136, 7784}, {264, 683}, {371, 8946}, {372, 8948}, {427, 6340}, {460, 6524}, {1593, 9737}, {1692, 2207}, {1722, 8769}, {5139, 13881}, {7745, 9777}, {8743, 8753}, {8754, 10602}, {9732, 11394}, {9733, 11395}

X(14248) = isogonal conjugate of X(6337)
X(14248) = X(i)-cross conjugate of X(j) for these (i,j): {6, 25}, {512, 3565}, {1196, 393}
X(14248) = cevapoint of X(i) and X(j) for these (i,j): {6, 8770}, {512, 2971}, {8754, 12075}
X(14248) = trilinear pole of line {2489, 8651}
X(14248) = crosssum of X(i) and X(j) for these (i,j): {3, 6461}, {3167, 10607}
X(14248) = X(371)-vertex conjugate of X(372)
X(14248) = Cundy-Parry Phi transform of X(3563)
X(14248) = Cundy-Parry Psi transform of X(3564)
X(14248) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4, 2996, 5203)
X(14248) = X(i)-isoconjugate of X(j) for these (i,j): {1, 6337}, {63, 193}, {69, 1707}, {75, 3167}, {92, 10607}, {304, 3053}, {326, 6353}, {1332, 3798}, {1444, 4028}, {3566, 4592}, {6091, 14210}
X(14248) = barycentric product X(i)*X(j) for these {i,j}: {4, 8770}, {19, 8769}, {25, 2996}, {111, 5203}, {393, 6391}, {2207, 6340}, {2501, 3565}
X(14248) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 6337}, {25, 193}, {32, 3167}, {184, 10607}, {1973, 1707}, {1974, 3053}, {2207, 6353}, {2333, 4028}, {2489, 3566}, {2971, 6388}, {2996, 305}, {3565, 4563}, {5203, 3266}, {6391, 3926}, {8769, 304}, {8770, 69}

X(14249) = X(3)X(107)∩X(4)X(51)

Trilinears    (sec^2 A)(cos A - cos B cos C) : :
Barycentrics    (SA*SB + SA*SC - SB*SC)/(a^2 SA^2) : : (Paul Yiu, Hyacinthos #21973 4/17/2013)
Barycentrics    b^2*c^2*(-a^2 + b^2 - c^2)^2*(a^2 + b^2 - c^2)^2*(-3*a^4 + 2*a^2*b^2 + b^4 + 2*a^2*c^2 - 2*b^2*c^2 + c^4) : :
X(14249) = 5 X(3091) - 2 X(8798)

The trilinear polar of X(12249) passes through X(6587). (Randy Hutson, November 2, 2017)

Let DEF be the Euler triangle of ABC. Let La be the line through D perpendicular to OD, and define Lb, Lc, cyclically. Let A'B'C' be the triangle formed by the lines La, Lb, Lc. The point X(14249) is the unique finite fixed point of the affine transformation that maps the reference triangle ABC onto A'B'C'. (Angel Montesdeoca, July 5, 2022)

X(14249) lies on the cubics K028, K412, K583, K647 and these lines: {2, 3346}, {3, 107}, {4, 51}, {5, 14057}, {20, 6525}, {76, 6528}, {92, 946}, {158, 273}, {235, 6530}, {253, 264}, {275, 10982}, {318, 8806}, {324, 3832}, {436, 11424}, {450, 13346}, {648, 11441}, {1249, 6616}, {1559, 2883}, {1598, 1629}, {1895, 5930}, {1941, 9306}, {3089, 11547}, {3542, 14165}, {4240, 11449}, {5895, 10152}, {6529, 8743}, {8884, 10594}

X(14249) = midpoint of X(4) and X(1075)
X(14249) = reflection of X(14059) in X(5)
X(14249) = isogonal conjugate of X(14379)
X(14249) = isotomic conjugate of X(15394)
X(14249) = X(264)-Ceva conjugate of X(2052)
X(14249) = cevapoint of X(i) and X(j) for these (i,j): {4, 6523}, {1249, 6525}
X(14249) = polar conjugate of X(1073)
X(14249) = Cundy-Parry Phi transform of X(1294)
X(14249) = Cundy-Parry Psi transform of X(6000)
X(14249) = X(14059)-of-Johnson-triangle
X(14249) = X(i)-cross conjugate of X(j) for these (i,j): {1559, 10152}, {2883, 20}, {8057, 107}
X(14249) = X(i)-isoconjugate of X(j) for these (i,j): {48, 1073}, {64, 255}, {394, 2155}, {459, 4100}, {577, 2184}, {2169, 8798}, {6056, 8809}
X(14249) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4, 1093, 2052), (4, 3168, 185), (6526, 10002, 3091)
X(14249) = barycentric product X(i)*X(j) for these {i,j}: {20, 2052}, {76, 6525}, {92, 1895}, {204, 1969}, {264, 1249}, {6528, 6587}
X(14249) = barycentric quotient X(i)/X(j) for these {i,j}: {4, 1073}, {20, 394}, {53, 8798}, {154, 577}, {158, 2184}, {204, 48}, {393, 64}, {610, 255}, {1093, 459}, {1096, 2155}, {1249, 3}, {1394, 7125}, {1562, 2972}, {1895, 63}, {1990, 11589}, {2052, 253}, {2883, 6509}, {3172, 184}, {3198, 3990}, {3213, 603}, {6523, 3343}, {6525, 6}, {6529, 1301}, {6587, 520}, {6616, 6617}, {7070, 2289}, {7156, 212}, {8804, 3682}, {13450, 13157}

X(14250) = X(3)X(7953)∩X(4)X(3096)

Barycentrics    a^2*(a^2 + 2*b^2 + c^2)*(a^2 + b^2 + 2*c^2)*(a^4 - b^4 - 3*b^2*c^2 - c^4) : :

X(14250) lies on the cubic K028 and these lines: {3, 7953}, {4, 3096}

. X(14250) = isogonal conjugate of X(14381)

X(14251) = X(3)X(805)∩X(4)X(147)

Barycentrics    a^4*(-b^2 + a*c)*(b^2 + a*c)*(a*b - c^2)*(a*b + c^2)*(a^2*b^2 - b^4 + a^2*c^2 - c^4) : :

X(14251) lies on the cubics K028, K166, K444, K516, K791, and these lines: {3, 805}, {4, 147}, {32, 8789}, {39, 512}, {182, 9467}, {263, 694}, {446, 511}, {9292, 9475}

X(14251) = isogonal conjugate of X(14382)
X(14251) = X(9468)-Ceva conjugate of X(237)
X(14251) = X(11672)-cross conjugate of X(237)
X(14251) = perspector of ABC and X(3511)-Brocard triangle
X(14251) = Cundy-Parry Phi transform of X(2698)
X(14251) = Cundy-Parry Psi transform of X(2782)
X(14251) = X(i)-isoconjugate for these (i,j): {98, 1966}, {290, 1580}, {336, 419}, {385, 1821}, {1910, 3978}, {1926, 1976}
X(14251) = barycentric product X(i)*X(j) for these {i,j}: {237, 1916}, {325, 9468}, {511, 694}, {805, 3569}, {881, 2396}, {882, 2421}, {1581, 1755}, {1934, 9417}, {1959, 1967}
X(14251) = barycentric quotient X(i)/X(j) for these {i,j}: {237, 385}, {511, 3978}, {694, 290}, {881, 2395}, {1755, 1966}, {1927, 1910}, {1959, 1926}, {1967, 1821}, {2211, 419}, {2421, 880}, {2491, 804}, {3289, 12215}, {8789, 1976}, {9417, 1580}, {9418, 1691}, {9468, 98}, {11672, 5976}

X(14252) =  (name pending)

Barycentrics    a^2*(a^2*b^2 - b^4 + 2*a^2*c^2 + b^2*c^2)*(2*a^2*b^2 + a^2*c^2 + b^2*c^2 - c^4)*(2*a^4*b^2 - 2*a^2*b^4 + 2*a^4*c^2 - a^2*b^2*c^2 - b^4*c^2 - 2*a^2*c^4 - b^2*c^4) : :

X(14252) lies on the cubic K028 and this line: {4, 39}

X(14252) = isogonal conjugate of X(14383)

X(14253) =  X(3)X(10425)∩X(4)X(99)

Barycentrics    a^2*(a^4 - a^2*b^2 + 2*b^4 - 2*a^2*c^2 - b^2*c^2 + c^4)*(a^4 - 2*a^2*b^2 + b^4 - a^2*c^2 - b^2*c^2 + 2*c^4)*(a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6 - 3*a^4*c^2 + a^2*b^2*c^2 + 3*a^2*c^4 - c^6) : :

X(14253) lies on the cubic K028 and these lines: {3, 10425}, {4, 99}, {249, 3053}

X(14253) = isogonal conjugate of X(14384)

X(14254) =  X(3)X(476)∩X(4)X(94)

Barycentrics    b^2*c^2*(a^2 - a*b + b^2 - c^2)*(a^2 + a*b + b^2 - c^2)*(-a^2 + b^2 - a*c - c^2)*(-a^2 + b^2 + a*c - c^2)*(-2*a^4 + a^2*b^2 + b^4 + a^2*c^2 - 2*b^2*c^2 + c^4) : :

X(14254) lies on the cubics K028, K054, K669 and these lines: {2, 1138}, {3, 476}, {4, 94}, {5, 523}, {13, 11085}, {14, 11080}, {76, 5641}, {381, 5627}, {382, 10688}, {1141, 11815}, {1989, 3767}, {3471, 10272}, {6644, 12028}, {6662, 10224}

X(14254) = isogonal conjugate of X(14385)
X(14254) = perspector of ABC and X(399)-Brocard triangle
X(14254) = Cundy-Parry Phi transform of X(477)
X(14254) = Cundy-Parry Psi transform of X(5663)
X(14254) = cevapoint of X(30) and X(10272)
X(14254) = X(i)-cross conjugate of X(j) for these (i,j): {113, 30}, {9033, 476}
X(14254) = X(i)-isoconjugate of X(i) for these (i,j): {50, 2349}, {74, 6149}, {323, 2159}
X(14254) = barycentric product X(i)*X(j) for these {i,j}: {30, 94}, {328, 1990}, {1989, 3260}, {2166, 14206}, {2407, 10412}, {6344, 11064}
X(14254) = barycentric quotient X(i)/X(j) for these {i,j}: {30, 323}, {94, 1494}, {1495, 50}, {1637, 526}, {1989, 74}, {1990, 186}, {2166, 2349}, {2173, 6149}, {2407, 10411}, {3163, 1511}, {3260, 7799}, {9033, 8552}, {10412, 2394}

X(14255) =  (name pending)

Barycentrics    b^2*c^2*(a^4 + 5*a^2*b^2 + b^4 - 4*a^2*c^2 - 4*b^2*c^2 + c^4)*(a^4 - 4*a^2*b^2 + b^4 + 5*a^2*c^2 - 4*b^2*c^2 + c^4)*(-8*a^4 + 5*a^2*b^2 + b^4 + 5*a^2*c^2 - 4*b^2*c^2 + c^4) : :

X(14255) lies on the cubic K028 and this line: {3, 9080}

barycentric quotient X(i)/X(j) for these {i,j}: {8598, 352}, {9080, 9190}, {9189, 9023}

X(14256) =  X(3)X(934)∩X(4)X(7)

Barycentrics    (a + b - c)^2*(a - b + c)^2*(a^3 + a^2*b - a*b^2 - b^3 + a^2*c - 2*a*b*c + b^2*c - a*c^2 + b*c^2 - c^3) : :

X(14256) lies on the cubic K028 and these lines: {2,2184}, {3,934}, {4,7}, {8,4566}, {40,347}, {57,279}, {76,4569}, {77,3345}, {85,189}, {142,10004}, {269,937}, {329,10402}, {348,658}, {387,3339}, {1020,3730}, {1111,11023}, {1375,5435}, {1441,11024}, {1467,3598}, {1817,6611}, {6904,9312}, {8074,8732}

X(14256) = isogonal conjugate of X(7367)
X(14256) = crosssum of X(41) and X(7118)
X(14256) = X(i)-cross conjugate of X(j) for these (i,j): {196, 7}, {223, 347}
X(14256) = X(i)-beth conjugate of X(j) for these (i,j): {85, 1847}, {664, 8}, {4573, 348}
X(14256) = X(i)-zayin conjugate of X(j) for these (i,j): {1, 7367}, {521, 657}
X(14256) = X(i)-Ceva conjugate of X(j) for these (i,j): {85, 279}, {7056, 7}
X(14256) = Cundy-Parry Phi transform of X(972)
X(14256) = Cundy-Parry Psi transform of X(971)
X(14256) = X(i)-isoconjugate of X(j) for these (i,j): {1, 7367}, {8, 7118}, {9, 2192}, {33, 268}, {41, 280}, {55, 282}, {78, 7154}, {84, 220}, {189, 1253}, {200, 1436}, {212, 7003}, {219, 7008}, {271, 607}, {281, 2188}, {285, 1334}, {346, 2208}, {480, 1422}, {657, 13138}, {728, 1413}, {1256, 7368}, {1260, 7129}, {1433, 7079}, {1440, 6602}, {1903, 2328}, {2287, 2357}, {3692, 7151}, {4130, 8059}
X(14256) = barycentric product X(i)*X(j) for these {i,j}: {7, 347}, {40, 1088}, {76, 6611}, {77, 342}, {85, 223}, {196, 348}, {208, 7182}, {221, 6063}, {269, 322}, {273, 7013}, {279, 329}, {331, 7011}, {479, 7080}, {1446, 1817}, {3668, 8822}, {4569, 6129}, {4626, 8058}, {7056, 7952}
X(14256) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 7367}, {7, 280}, {34, 7008}, {40, 200}, {56, 2192}, {57, 282}, {77, 271}, {196, 281}, {198, 220}, {208, 33}, {221, 55}, {222, 268}, {223, 9}, {227, 210}, {269, 84}, {273, 7020}, {278, 7003}, {279, 189}, {322, 341}, {329, 346}, {342, 318}, {347, 8}, {479, 1440}, {603, 2188}, {604, 7118}, {608, 7154}, {738, 1422}, {934, 13138}, {1014, 285}, {1042, 2357}, {1088, 309}, {1106, 2208}, {1398, 7151}, {1407, 1436}, {1427, 1903}, {1435, 7129}, {1817, 2287}, {2187, 1253}, {2199, 41}, {2324, 728}, {2331, 7079}, {2360, 2328}, {3194, 4183}, {3195, 7071}, {3209, 607}, {6129, 3900}, {6611, 6}, {6614, 8059}, {7011, 219}, {7013, 78}, {7023, 1413}, {7053, 1433}, {7074, 480}, {7078, 1260}, {7080, 5423}, {7114, 212}, {7952, 7046}, {8058, 4163}, {8822, 1043}

X(14257) =  X(3)X(108)∩X(4)X(65)

Barycentrics    (a + b - c)*(a - b + c)*(a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*(a^4 - b^4 + 2*a^2*b*c - 2*a*b^2*c - 2*a*b*c^2 + 2*b^2*c^2 - c^4) : :

X(14257) lies on the cubic K028 and these lines: {3,108}, {4,65}, {7,2995}, {34,998}, {46,208}, {56,1068}, {57,225}, {278,961}, {318,377}, {406,1940}, {860,1788}, {1249,2182}, {1895,6836}, {3485,5136}, {5794,7046}, {7412,11507}, {7414,11509}
X(14257) = isogonal conjugate of X(39167)
X(14257) = cevapoint of X(478) and X(17408)
X(14257) = crosspoint of X(18026) and X(23984)
X(14257) = crosssum of X(1946) and X(35072)
X(14257) = X(264)-Ceva conjugate of X(278)
X(14257) = polar conjugate of X(34277)
X(14257) = X(1897)-beth conjugate of X(8)
X(14257) = Cundy-Parry Phi transform of X(1295)
X(14257) = Cundy-Parry Psi transform of X(6001)
X(14257) = X(i)-isoconjugate of X(j) for these (i,j): {78, 3435}, {212, 8048}
X(14257) = barycentric product X(i)*X(j) for these {i,j}: {197, 331}, {264, 478}, {273, 1766}, {278, 3436}
X(14257) = barycentric quotient X(i)/X(j) for these {i,j}: {197, 219}, {205, 212}, {278, 8048}, {478, 3}, {608, 3435}, {1766, 78}, {3436, 345}, {6588, 521}
X(14257) = trilinear product X(i)*X(j) for these {i,j}: {4, 21147}, {34, 3436}, {75, 17408}, {92, 478}, {108, 21186}, {123, 24033}, {197, 273}, {205, 331}, {225, 16049}, {278, 1766}, {608, 20928}, {653, 6588}, {1396, 21074}, {5307, 34263}

X(14258) =  X(3)X(1310)∩X(4)X(75)

Barycentrics    (a^2 + 2*a*b + b^2 + c^2)*(a^2 + b^2 + 2*a*c + c^2)*(a^3 + a^2*b - a*b^2 - b^3 + a^2*c - 2*a*b*c - b^2*c - a*c^2 - b*c^2 - c^3) : :

X(14258) lies on the cubic K028 and these lines: {3,1310}, {4,75}, {81,2221}
X(14258) = barycentric quotient X(i)/X(j) for these {i,j}: {406, 7102}, {5739, 2345}, {12514, 612}

X(14259) =  X(3)X(907)∩X(4)X(141)

Barycentrics    a^2*(a^2 + 3*b^2 + c^2)*(a^2 + b^2 + 3*c^2)*(a^4 - b^4 - 4*b^2*c^2 - c^4) : :

X(14259) lies on the cubic K028 and these lines: {3,907}, {4,141}, {3108,5359}

X(14259) = barycentric quotient X(i)/X(j) for these {i,j}: {7485, 3618}

X(14260) =  X(3)X(901)∩X(4)X(145)

Barycentrics    a^2*(a + b - 2*c)*(a - 2*b + c)*(a^2*b - b^3 + a^2*c - 2*a*b*c + b^2*c + b*c^2 - c^3) : :

Let P be the point on line X(3)X(8) for which the P-Brocard triangle is perspective to ABC. P lies on the rectangular hyperbola {{X(1),X(3),X(6),X(399),X(2930),X(2948),X(3511)}}, centered at X(110). The perspector of ABC and the P-Brocard triangle is X(14260). (Randy Hutson, November 17, 2019)

X(14260) lies on the cubics K028, K165, K360, K594 and thesse lines:
{1,513}, {3,901}, {4,145}, {56,106}, {76,4555}, {88,957}, {218,2316}, {945,11248}, {956,3257}, {999,1318}, {1168,2099}, {2098,3326}, {2183,2427}, {3216,4674}, {5730,6790}

X(14260) = isogonal conjugate of X(36944)
X(14260) = cevapoint of X(i) and X(j) for these {i,j}: {1361, 1457}, {8677, 35012}
X(14260) = crosspoint of X(106) and X(1168)
X(14260) = crossdifference of every pair of points on line {44, 1639}
X(14260) = crosssum of X(i) and X(j) for these (i,j): {214, 519}, {1960, 35092}, {3689, 4370}
X(14260) = X(i)-Ceva conjugate of X(j) for these (i,j): {1318, 106}, {4555, 10015}
X(14260) = X(i)-cross conjugate of X(j) for these (i,j): {1457, 106}, {8677, 901}
X(14260) = trilinear pole of line {2183, 3310}
X(14260) = X(2183)-zayin conjugate of X(44)
X(14260) = Cundy-Parry Phi transform of X(953)
X(14260) = Cundy-Parry Psi transform of X(952)
X(14260) = X(i)-isoconjugate of X(j) for these (i,j): {104, 519}, {909, 4358}, {1023, 2401}, {1635, 13136}, {1809, 1877}, {2720, 4768}, {4738, 10428}
X(14260) = barycentric product X(i)*X(j) for these {i,j}: {88, 517}, {106, 908}, {859, 4080}, {901, 10015}, {903, 2183}, {1145, 2226}, {1320, 1465}, {1457, 4997}, {1769, 3257}, {1785, 1797}, {2427, 6548}, {3262, 9456}, {3310, 4555}, {3929, 5556}
X(14260) = barycentric quotient X(i)/X(j) for these {i,j}: {517, 4358}, {901, 13136}, {908, 3264}, {1457, 3911}, {1769, 3762}, {2183, 519}, {3310, 900}, {9456, 104}

X(14261) =  X(3)X(106)∩X(4)X(519)

Barycentrics    a^2*(a + b - 3*c)*(a - 3*b + c)*(a^2*b - b^3 + a^2*c - 3*a*b*c + 2*b^2*c + 2*b*c^2 - c^3) : :

X(14261) lies on the cubic K028 and these lines: {3,106}, {4,519}, {2051,6557}, {2098,4014}, {3091,6556}, {4373,10446}, {6048,10563}, {7991,8056}, {9050,11477}

X(14261) = Cundy-Parry Phi transform of X(106)
X(14261) = Cundy-Parry Psi transform of X(519)
X(14261) = barycentric product X(i)*X(j) for these {i,j}: {4052, 7419}

X(14262) =  X(3)X(111)∩X(4)X(524)

Barycentrics    a^2*(a^2 + b^2 - 5*c^2)*(a^2 - 5*b^2 + c^2)*(a^4 - b^4 + 4*b^2*c^2 - c^4) : :

Let P be a finite point in the plane of ABC. Let (Oa) be the circumcircle of BCP, and define (Ob) and (Oc) cyclically. Let A' be the intersection, other than P, of (Oa) and AP, and define B' and C' cyclically. Let A" be the antipode of A' in (Oa), and define B" and C" cyclically. The lines AA", BB" CC" concur for all P. When P = X(6), the lines AA", BB" CC" concur in X(14262). (Randy Hutson, November 2, 2017)

X(14262) lies on the cubics K028 and X272 and on these lines: {3,111}, {4,524}, {6,9871}, {1975,2418}, {8667,13168}
X(14262) = reflection of X(1296) in X(10354)
X(14262) = isogonal conjugate of X(13608)
X(14262) = antigonal conjugate of X(34166)
X(14262) = X(671)-Ceva conjugate of X(13492)
X(14262) = X(8542)-cross conjugate of X(1995)
X(14262) = syngonal conjugate of X(10354)
X(14262) = Cundy-Parry Phi transform of X(111)
X(14262) = Cundy-Parry Psi transform of X(524)
X(14262) = barycentric product X(i)*X(j) for these {i,j}: {1995, 5485}
X(14262) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 13608}, {1995, 1992}, {8542, 11165}, {11185, 11059}

X(14263) =  X(3)X(111)∩X(4)X(1499)

Barycentrics    a^2*(a^2 + b^2 - 2*c^2)*(a^2 - 2*b^2 + c^2)*(a^2*b^2 + b^4 + a^2*c^2 - 4*b^2*c^2 + c^4)

X(14263) lies on the cubic K028 and these lines: {3, 111}, {4, 1499}, {6, 10630}, {39, 5968}, {76, 338}, {691, 3053}, {892, 7754}, {3291, 11634}, {5286, 9214}, {5359, 8877}, {8743, 8753}

X(14263) = isogonal conjugate of X(34161)
X(14263) = X(8681)-cross conjugate of X(111)
X(14263) = crosspoint of X(671) and X(10630)
X(14263) = crossdifference of every pair of points on line {3292, 9125}
X(14263) = crosspoint of X(671) and X(10630)
X(14263) = crosssum of X(i) and X(j) for these (i,j): {187, 2482}, {14417, 23992}
X(14263) = Cundy-Parry Phi transform of X(1296)
X(14263) = Cundy-Parry Psi transform of X(1499)
X(14263) = barycentric product X(i)*X(j) for these {i,j}: {126, 10630}, {671, 3291}, {691, 9134}, {5466, 11634}
X(14263) = barycentric quotient X(i)/X(j) for these {i,j}: {3291, 524}, {5140, 468}, {8681, 6390}, {8753, 2374}, {11634, 5468}

X(14264) =  X(3)X(74)∩X(4)X(523)

Barycentrics    a^2*(a^4 - 2*a^2*b^2 + b^4 + a^2*c^2 + b^2*c^2 - 2*c^4)*(a^4 + a^2*b^2 - 2*b^4 - 2*a^2*c^2 + b^2*c^2 + c^4)*(a^4*b^2 - 2*a^2*b^4 + b^6 + a^4*c^2 + 2*a^2*b^2*c^2 - b^4*c^2 - 2*a^2*c^4 - b^2*c^4 + c^6)

X(14264) lies on the cubics K028, K389, K568, K622, and these lines: {3, 74}, {4, 523}, {5, 12079}, {6, 11074}, {24, 1304}, {76, 1494}, {186, 5502}, {378, 10419}, {381, 5627}, {1624, 7722}, {2132, 3440}, {8743, 8749}

X(14264) = X(1494)-Ceva conjugate of X(3580)
X(14264) = X(13754)-cross conjugate of X(74)
X(14264) = X(i)-isoconjugate of X(j) for these (i,j): {687, 2631}, {1099, 10419}, {1784, 5504}, {2173, 2986}
X(14264) = crosspoint of X(i) and X(j) for these (i,j): {74, 5627}
X(14264) = trilinear pole of line {686, 3003}
X(14264) = crossdifference of every pair of points on line {1637, 3284}
X(14264) = crosssum of X(i) and X(j) for these (i,j): {30, 1511}, {1495, 3163}, {3258, 9033}
X(14264) = Cundy-Parry Phi transform of X(110)
X(14264) = Cundy-Parry Psi transform of X(523)
X(14264) = barycentric product X(i)*X(j) for these {i,j}: {74, 3580}, {1304, 6334}, {1494, 3003}, {1725, 2349}
X(14264) = barycentric quotient X(i)/X(j) for these {i,j}: {74, 2986}, {686, 9033}, {1304, 687}, {1725, 14206}, {3003, 30}, {3580, 3260}, {8749, 1300}, {11079, 12028}, {13754, 11064}

X(14265) =  X(3)X(76)∩X(4)X(512)

Barycentrics    b^2*c^2*(a^4 + b^4 - a^2*c^2 - b^2*c^2)*(-a^4 + a^2*b^2 + b^2*c^2 - c^4)*(2*a^4 - a^2*b^2 + b^4 - a^2*c^2 - 2*b^2*c^2 + c^4) : :

X(14265) lies on the cubic K028 and these lines: {3, 76}, {4, 512}, {264, 2065}, {575, 5967}, {847, 6531}, {2966, 6179}

X(14265) = isogonal conjugate of X(34157)
X(14265) = X(3564)-cross conjugate of X(98)
X(14265) = cevapoint of X(2974) and X(3564)
X(14265) = crossdifference of every pair of points on line {2491, 3289}
X(14265) = crosssum of X(237) and X(11672)
X(14265) = Cundy-Parry Phi transform of X(99)
X(14265) = Cundy-Parry Psi transform of X(512)
X(14265) = X(i)-isoconjugate of X(j) for these (i,j): {237, 8773}, {1755, 2987}, {8781, 9417}
X(14265) = X(290)-daleth conjugate of X(98)
X(14265) = barycentric product X(i)*X(j) for these {i,j}: {230, 290}, {1733, 1821}
X(14265) = barycentric quotient X(i)/X(j) for these {i,j}: {98, 2987}, {230, 511}, {290, 8781}, {460, 232}, {1692, 237}, {1733, 1959}, {1821, 8773}, {2966, 10425}, {4226, 2421}, {5477, 9155}, {6531, 3563}, {8772, 1755}

X(14266) =  X(3)X(8)∩X(4)X(513)

Barycentrics    (a^3 - a^2*b - a*b^2 + b^3 + 2*a*b*c - a*c^2 - b*c^2)*(a^3 - a*b^2 - a^2*c + 2*a*b*c - b^2*c - a*c^2 + c^3)*(a^3*b - a^2*b^2 - a*b^3 + b^4 + a^3*c + a*b^2*c - a^2*c^2 + a*b*c^2 - 2*b^2*c^2 - a*c^3 + c^4) : :

X(14266) lies on the cubic K028 and these lines: {3, 8}, {4, 513}, {499, 10571}, {1479, 10777}, {2250, 3730}

X(14266) = isogonal conjugate of X(39173)
X(14266) = cevapoint of X(i) and X(j) for these {i,j}: {912, 34332}, {1737, 11570}
X(14266) = X(i)-cross conjugate of X(j) for these (i,j): {912, 104}, {12832, 1737}
X(14266) = X(i)-isoconjugate X(j) for these (i,j): {1769, 6099}, {2183, 2990}
X(14266) = X(1309)-beth conjugate of X(1068)
X(14266) = X(2323)-zayin conjugate of X(2183)
X(14266) = Cundy-Parry Phi transform of X(100)
X(14266) = Cundy-Parry Psi transform of X(513)
X(14266) = barycentric quotient X(i)/X(j) for these {i,j}: {104, 2990}, {1737, 908}, {8609, 517}
X(14266) = trilinear product X(i)*X(j) for these {i,j}: {104, 1737}, {912, 36123}, {2252, 16082}, {8609, 34234}

X(14267) =  X(3)X(105)∩X(4)X(885)

Barycentrics    (a^2 + b^2 - a*c - b*c)*(a^2 - a*b - b*c + c^2)*(a^2*b + b^3 + a^2*c - 2*a*b*c - b^2*c - b*c^2 + c^3) : :

X(14267) lies on the cubic K028 and these lines: {3, 105}, {4, 885}, {8, 76}, {673, 5698}, {5091, 5222}, {8743, 8751}

X(14267) = isogonal conjugate of X(34159)
X(14267) = trilinear pole of line X(3290)X(23770)
X(14267) = crosspoint of X(2481) and X(6185)
X(14267) = cevapoint of X(3290) and X(20455)
X(14267) = crosssum of X(2223) and X(6184)
X(14267) = crosspoint of X(2481) and X(6185)
X(14267) = crosssum of X(2223) and X(6184)
X(14267) = X(3684)-zayin conjugate of X(672)
X(14267) = Cundy-Parry Phi transform of X(1292)
X(14267) = Cundy-Parry Psi transform of X(3309)
X(14267) = X(672)-isoconjugate of X(2991)
X(14267) = barycentric product X(i)*X(j) for these {i,j}: {120, 6185}, {673, 1738}, {2481, 3290}
X(14267) = barycentric quotient X(i)/X(j) for these {i,j}: {105, 2991}, {120, 4437}, {1738, 3912}, {3290, 518}

X(14268) =  X(3)X(105)∩X(4)X(518)

Barycentrics    (a^2 - 2*a*b + b^2 - 2*b*c + c^2)*(a^2 + b^2 - 2*a*c - 2*b*c + c^2)*(a^3 - a^2*b + a*b^2 - b^3 - a^2*c + b^2*c + a*c^2 + b*c^2 - c^3) : :

X(14268) lies on the cubic K028 and these lines: {3, 105}, {4, 518}, {991, 2191}, {7397, 7964}

X(14268) = Cundy-Parry Phi transform of X(105)
X(14268) = Cundy-Parry Psi transform of X(518)
X(14268) = X(3433)-isoconjugate of X(3870)
X(14268) = barycentric product X(277)X(3434)
X(14268) = barycentric quotient X(i)/X(j) for these {i,j}: {169, 3870}, {277, 13577}, {1486, 218}, {3434, 344}, {5452, 6600}

X(14269) =  X(13)X(12821)∩X(14)X(12820)

Barycentrics    7 a^4+a^2 b^2-8 b^4+a^2 c^2+16 b^2 c^2-8 c^4 : :
X(14269) = 8 X(2)-5 X(3) = X(2)+5 X(4) = X(3)+8 X(4) = 7 X(3)-16 X(5) = 7 X(2)-10 X(5) = 7 X(4)+2 X(5) = 17 X(3)-8 X(20) = 17 X(2)-5 X(20) = 17 X(4)+X(20) = 11 X(20)-17 X(376) = 11 X(3)-8 X(376) = 11 X(2)-5 X(376) = 11 X(4)+X(376) = 2 X(20)-17 X(381) = 2 X(376)-11 X(381) = 4 X(5)-7 X(381) = 2 X(2)-5 X(381) = X(3)-4 X(381) = 2 X(4)+X(381) = 10 X(4)-X(382) = 2 X(2)+X(382) = 5 X(381)+X(382) = 5 X(3)+4 X(382) = 10 X(376)+11 X(382) = 10 X(20)+17 X(382) = 5 X(5)-14 X(546) = 5 X(381)-8 X(546) = X(2)-4 X(546) = 5 X(4)+4 X(546) = X(382)+8 X(546) = 17 X(5)-14 X(547) = 17 X(381)-8 X(547) = 17 X(546)-5 X(547) = X(20)-4 X(547) = 17 X(4)+4 X(547) = 13 X(3)-16 X(549) = 13 X(2)-10 X(549) = 13 X(5)-7 X(549) = 13 X(381)-4 X(549) = 13 X(4)+2 X(549) = 5 X(2)-2 X(550) = 10 X(546)-X(550) = 5 X(382)+4 X(550) = 2 X(376)-5 X(1656) = 11 X(381)-5 X(1656) = 13 X(3)-4 X(1657) = 13 X(381)-X(1657) = 4 X(549)-X(1657) = 13 X(382)+5 X(1657) = 13 X(381)-10 X(3091) = X(1657)-10 X(3091) = 2 X(549)-5 X(3091) = 13 X(4)+5 X(3091) = 19 X(382)-10 X(3146) = 19 X(4)-X(3146) = 19 X(381)+2 X(3146) = 19 X(2)+5 X(3146) = 19 X(3)+8 X(3146) = 19 X(376)+11 X(3146) = 19 X(20)+17 X(3146) = 14 X(5)-5 X(3522) = 7 X(20)-17 X(3524)

X(14269) is related to points on the asymptotes of the cubic K028.

X(14269) lies on these lines: {2,3}, {13,12821}, {14,12820}, {143,11439}, {265,3531}, {542,5093}, {590,9690}, {1327,3071}, {1328,3070}, {1539,9140}, {1699,10247}, {3058,9654}, {3244,3656}, {3582,12943}, {3583,6767}, {3584,12953}, {3585,7373}, {3626,12699}, {3632,8148}, {3636,3655}, {3653,3817}, {3715,12702}, {3982,5722}, {5050,11645}, {5339,12817}, {5340,12816}, {5434,9669}, {5655,12902}, {5663,13321}, {5690,10248}, {5946,11455}, {6033,9880}, {6054,12355}, {6199,6564}, {6329,11179}, {6395,6565}, {6474,8981}, {6475,13966}, {7999,11017}, {9605,11648}, {9655,10072}, {9668,10056}, {9704,11424}, {9730,13570}, {10095,12290}, {10113,10706}, {11008,11180}

X(14269) = midpoint of X(i) and X(j) for these {i,j}: {4, 3839}, {3524, 3543}, {3627, 11539}, {3830, 5055}
X(14269) = reflection of X(i) in X(j) for these {i,j}: {3, 5055}, {376, 11539}, {381, 3839}, {3524, 5}, {3534, 3524}, {3653, 3817}, {3839, 3845}, {5054, 3545}, {5055, 381}, {11539, 5066}
X(14269) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 376, 3530), (2, 381, 3851), (2, 546, 381), (2, 3528, 549), (2, 3543, 3529), (2, 3855, 11737), (2, 11737, 5079), (4, 5, 5076), (4, 20, 12102), (4, 381, 3830), (4, 546, 382), (4, 3091, 3853), (4, 3543, 12101), (4, 3832, 3627), (4, 3843, 3), (4, 3845, 381), (5, 3543, 3534), (5, 5073, 3), (5, 5076, 5073), (5, 10303, 1656), (5, 12101, 3543), (20, 3858, 5072), (140, 11001, 14093), (376, 3832, 5066), (376, 5066, 1656), (381, 382, 2), (381, 1656, 5066), (381, 3534, 5), (381, 3830, 3), (381, 3845, 3843), (381, 5054, 3545), (381, 5076, 3534), (381, 12101, 5073), (382, 546, 3851), (382, 3529, 5073), (382, 3851, 3), (382, 5079, 550), (546, 550, 3855), (549, 3860, 3091), (550, 3855, 5079), (550, 11737, 2), (1593, 13621, 3), (1657, 3091, 5070), (1657, 5070, 3), (2043, 2044, 3628), (3091, 3533, 5), (3091, 3853, 1657), (3091, 3860, 381), (3146, 3850, 3526), (3146, 5071, 8703), (3528, 3853, 382), (3529, 3533, 3528), (3534, 3543, 5073), (3534, 5076, 3543), (3534, 12101, 3830), (3543, 5076, 3830), (3543, 12101, 5076), (3545, 5054, 5055), (3627, 3832, 1656), (3627, 5066, 376), (3830, 3843, 381), (3830, 5073, 3543), (3832, 5066, 381), (3843, 3851, 546), (3850, 8703, 5071), (3853, 3860, 549), (3855, 5079, 3851), (3858, 12102, 20), (5071, 8703, 3526), (5071, 10299, 2), (5899, 9818, 3), (11317, 14041, 11286)

X(14270) =  CROSSPOINT OF X(6) AND X(476)

Trilinears    (sin^2 A)(1 + 2 cos 2A)sin(B - C) : :
Trilinears    (sin A)(sin B csc 3B - sin C csc 3C) : :
Barycentrics    a^4 (b^2-c^2) (a^2-b^2-b c-c^2) (a^2-b^2+b c-c^2) : :
X(14270) = 3 X(351) - 2 X(6140) = X(8552) - 3 X(9126) = 3 X(351) + X(9409) = 2 X(6140) + X(9409) = X(887) + 3 X(9420)

X(14270) is related to the cubic K009.

X(14270) lies on these lines: {3,690}, {23,9185}, {32,2491}, {110,9160}, {182,7638}, {187,237}, {523,5926}, {526,1511}, {804,12042}, {878,3455}, {888,9142}, {1995,9189}, {2079,7669}, {2931,9033}, {5961,13289}, {6132,9517}, {7496,9191}, {9003,12584}.

X(14270) = isogonal conjugate of X(35139)
X(14270) = X(i)-Ceva conjugate of X(j) for these (i,j): {186, 2088}, {476, 6}, {1141, 115}, {2433, 3049}
X(14270) = crosspoint of X(6) and X(476)
X(14270) = crosssum of X(i) and X(j) for these (i,j): {2, 526}, {94, 10412}, {523, 3580}, {525, 2072}, {690, 13162}, {850, 3260}
X(14270) = crossdifference of every pair of points on line {2, 94}
X(14270) = circumcircle-pole of the Fermat axis
X(14270) = isogonal conjugate of the isotomic conjugate of X(526)
X(14270) = {X(351),X(5191)}-harmonic conjugate of X(5027)
X(14270) = center of circle that is the inverse-in-circumcircle of line X(2)X(98); see http://bernard-gibert.fr/Exemples/k009.html
X(14270) = perspector of conic {A,B,C,X(6),X(50)}; see http://bernard-gibert.fr/Exemples/k009.html
X(14270) = X(i)-isoconjugate of X(j) for these (i,j): {75, 476}, {94, 662}, {99, 2166}, {162, 328}, {265, 811}, {799, 1989}, {4592, 6344}, {4602, 11060}
X(14270) = X(6)-vertex conjugate of X(3016)
X(14270) = barycentric product X(i)X(j) for these {i,j}: {1, 2624}, {6, 526}, {15, 6138}, {16, 6137}, {25, 8552}, {32, 3268}, {50, 523}, {54, 2081}, {110, 2088}, {186, 647}, {187, 9213}, {323, 512}, {340, 3049}, {654, 2594}, {661, 6149}, {669, 7799}, {1154, 2623}, {1464, 9404}, {1511, 2433}, {1983, 2611}, {2245, 2605}, {2290, 2616}, {2436, 5663}, {3124, 10411}
X(14270) = barycentric quotient X(i)/X(j) for these {i,j}: {32, 476}, {50, 99}, {186, 6331}, {323, 670}, {512, 94}, {526, 76}, {647, 328}, {669, 1989}, {798, 2166}, {2081, 311}, {2088, 850}, {2489, 6344}, {2624, 75}, {3049, 265}, {3124, 10412}, {3268, 1502}, {6137, 301}, {6138, 300}, {6149, 799}, {7799, 4609}, {8552, 305}, {9126, 11059}, {9426, 11060}

X(14271) =  MIDPOINT OF X(182) AND X(14270)

Barycentrics    a^2 (b^2-c^2) (2 a^8-3 a^6 b^2+a^2 b^6-3 a^6 c^2-2 a^4 b^2 c^2+3 a^2 b^4 c^2+3 a^2 b^2 c^4-2 b^4 c^4+a^2 c^6) : :

X(14271) is related to the cubic K009.

X(14271) lies on these lines: {182,7638}, {512,11621}, {690,5092}, {1495,9189}, {1511,9003}, {1691,2491}, {8675,9126}

X(14271) = midpoint of X(182) and X(14270)

leftri

Triaxial points: X(14272)-X(14353)

rightri

This preamble and centers X(14272)-X(14353) were contributed by César Eliud Lozada, September 6, 2017.

"Let F1, F2, F3 be three figures in perspective two and two in the same plane, show that if they have a common centre of perspective, their three axes of perspective are concurrent." (Quoted from Lachlan, R.: An Elementary Treatise on Modern Pure Geometry, McMillan & Co., 1893, pp. 123).

For three triangles T1, T2, T3 satisfying the those conditions, the point of concurrence of the three axes is here named the triaxial point of the triangles.

Among thousands of triaxial points, centers X(14272)-X(1353) have been selected for this section.

X(14272) =  TRIAXIAL POINT OF THESE TRIANGLES: ABC, 4th BROCARD, CIRCUMMEDIAL

Barycentrics    (2*a^2-b^2-c^2)*(2*a^4+(b^2+c^2)*a^2-b^4+4*b^2*c^2-c^4)*(b^2-c^2) : :
X(14272) = 4*X(2492)-3*X(8371) = 2*X(6131)-3*X(9123) = 3*X(8371)-2*X(14277) = 3*X(9148)-2*X(14278)

X(14272) lies on these lines: {23,385}, {690,5095}, {2492,8371}, {6131,9123}, {9148,14278}, {9979,11631}

X(14272) = reflection of X(14277) in X(2492)
X(14272) = {X(2492), X(14277)}-harmonic conjugate of X(8371)

X(14273) =  TRIAXIAL POINT OF THESE TRIANGLES: ABC, 4th BROCARD, 5th EULER

Barycentrics    (2*a^2-b^2-c^2)*(b^2-c^2)*(a^2-b^2+c^2)*(a^2+b^2-c^2) : :
X(14273) = 3*X(1637)-4*X(2492) = 3*X(2489)-2*X(2501)

X(14273) lies on the cubic K150 and these lines: {4,2793}, {24,11616}, {99,112}, {111,2374}, {115,2971}, {126,1560}, {230,231}, {690,5095}, {804,5186}, {888,1843}, {1974,5027}, {2395,6531}, {2451,6368}, {3163,9475}, {3569,9033}, {4232,9123}, {5094,14277}, {5523,8430}, {6333,9035}, {7665,9979}, {8791,9178}, {14278,14279}

X(14273) = reflection of X(14278) in X(14279)
X(14273) = polar conjugate of X(892)
X(14273) = trilinear pole of the line {1648, 2682}
X(14273) = PU(4)-harmonic conjugate of X(10418)
X(14273) = pole wrt polar circle of trilinear polar of X(892) (line X(2)X(99))
X(14273) = X(63)-isoconjugate of X(691)
X(14273) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2492, 6131, 10418), (8105, 8106, 2489)

X(14274) =  TRIAXIAL POINT OF THESE TRIANGLES: ABC, 4th BROCARD, 3rd PARRY

Barycentrics    (SB^2-SC^2)*(3*SA-SW)*(9*S^2*(3*S^2-SW^2)+12*(3*S^2+SW^2)*SW*R^2-4*SW^4-4*(9*(3*S^2+SW^2)*R^2-2*SW^3)*SA+3*(SW^2+9*S^2)*SA^2)*((3*S^2+SW^2)*SA-2*S^2*SW) : :

X(14274) lies on these lines: {690,5095}, {2080,14275}

X(14275) =  TRIAXIAL POINT OF THESE TRIANGLES: ABC, CIRCUMMEDIAL, 3rd PARRY

Barycentrics    (SB^2-SC^2)*(3*S^2*(4*(3*S^2+SW^2)*R^2-(S^2+SW^2)*SW)+(S^2+SW^2)*(3*SA-2*SW)*SA*SW)*((3*S^2+SW^2)*SA-2*S^2*SW) : :

X(14275) lies on these lines: {23,385}, {2080,14274}

X(14276) =  TRIAXIAL POINT OF THESE TRIANGLES: ABC, 5th EULER, 3rd PARRY

Barycentrics    (SB^2-SC^2)*(6*(3*R^2-SW)*S^2+SW^2*(6*R^2-SW)-SA*SW*(3*SA-2*SW))*((3*S^2+SW^2)*SA-2*S^2*SW) : :

X(14276) lies on these lines: {230,231}, {2080,14274}

X(14277) =  TRIAXIAL POINT OF THESE TRIANGLES: ANTICOMPLEMENTARY, 4th BROCARD, CIRCUMMEDIAL

Barycentrics    (a^6+3*b^2*c^2*a^2+(b^2+c^2)*(b^4-4*b^2*c^2+c^4))*(b^2-c^2) : :
X(14277) = 2*X(2492)-3*X(8371) = 3*X(8371)-X(14272) = 3*X(9148)-2*X(14279)

X(14277) lies on these lines: {3,2793}, {67,690}, {325,523}, {338,3143}, {2492,8371}, {5094,14273}, {7495,9123}, {9134,9178}

X(14277) = isotomic conjugate of X(35569)
X(14277) = reflection of X(i) in X(j) for these (i,j): (9178, 9134), (14272, 2492)
X(14277) = {X(8371), X(14272)}-harmonic conjugate of X(2492)

X(14278) =  TRIAXIAL POINT OF THESE TRIANGLES: ANTICOMPLEMENTARY, 4th BROCARD, 5th EULER

Barycentrics    (4*a^6+3*(b^2+c^2)*a^4+6*b^2*c^2*a^2+(b^2+c^2)*(b^4-10*b^2*c^2+c^4))*(b^2-c^2) : :
X(14278) = 3*X(9148)-X(14272)

X(14278) lies on these lines: {67,690}, {523,2525}, {9148,14272}, {14273,14279}

X(14278) = reflection of X(14273) in X(14279)

X(14279) =  TRIAXIAL POINT OF THESE TRIANGLES: 4th BROCARD, 5th EULER, MEDIAL

Barycentrics    (a^6-3*(b^2+c^2)*a^4-3*(b^4-b^2*c^2+c^4)*a^2+(b^2+c^2)^3)*(b^2-c^2) : :
X(14279) = 3*X(9148)-X(14277)

X(14279) lies on these lines: {83,9180}, {325,523}, {690,2492}, {14273,14278}

X(14279) = midpoint of X(14273) and X(14278)
X(14279) = isotomic conjugate of X(35570)

X(14280) =  TRIAXIAL POINT OF THESE TRIANGLES: ABC, CONWAY, HUTSON EXTOUCH

Barycentrics    (b-c)*(a^3-3*(b+c)*a^2+3*(b-c)^2*a-(b^2-c^2)*(b-c))*(3*a^3-5*(b+c)*a^2+(b-c)^2*a+(b^2-c^2)*(b-c)) : :

X(14280) lies on the line {239,514}

X(14280) = reflection of X(14281) in X(14283)

X(14281) =  TRIAXIAL POINT OF THESE TRIANGLES: ABC, 2nd CONWAY, HUTSON EXTOUCH

Barycentrics    (b-c)*(a^3+(b+c)*a^2-(5*b^2+6*b*c+5*c^2)*a+3*(b^2-c^2)*(b-c))*(3*a^3-5*(b+c)*a^2+(b-c)^2*a+(b^2-c^2)*(b-c)) : :

X(14281) lies on the line {514,661}

X(14281) = reflection of X(14280) in X(14283)

X(14282) =  TRIAXIAL POINT OF THESE TRIANGLES: ABC, HONSBERGER, HUTSON EXTOUCH

Barycentrics    (b-c)*(-a+b+c)*(3*a^3-5*(b+c)*a^2+(b-c)^2*a+(b^2-c^2)*(b-c)) : :

X(14282) lies on these lines: {514,657}, {522,650}, {523,14330}, {14300,14303}

X(14283) =  TRIAXIAL POINT OF THESE TRIANGLES: ABC, HUTSON EXTOUCH, INTOUCH

Barycentrics    (b-c)*(-a+b+c)*((b+c)*a-(b-c)^2)*(3*a^3-5*(b+c)*a^2+(b-c)^2*a+(b^2-c^2)*(b-c)) : :

X(14283) lies on the line {241,514}

X(14283) = midpoint of X(14280) and X(14281)

X(14284) =  TRIAXIAL POINT OF THESE TRIANGLES: ABC, EXTOUCH, HUTSON INTOUCH

Barycentrics    (b-c)*(-a+b+c)*((b+c)*a+(b-c)^2)*(3*a-b-c) : :

X(14284) lies on these lines: {522,650}, {900,1459}, {3057,6363}, {3667,4162}, {4017,4854}, {4404,4918}

X(14285) =  TRIAXIAL POINT OF THESE TRIANGLES: ANTICOMPLEMENTARY, ATIK, FUHRMANN

Barycentrics    (b-c)*(6*a^4-3*(b+c)*a^3-(b^2-10*b*c+c^2)*a^2+(b+c)*(7*b^2-10*b*c+7*c^2)*a-(b+c)^4) : :

X(14285) lies on these lines: {80,900}, {513,2529}, {2827,14287}

X(14286) =  TRIAXIAL POINT OF THESE TRIANGLES: ANTICOMPLEMENTARY, 2nd CONWAY, FUHRMANN

Barycentrics    (b-c)*(a^4+5*b*c*a^2+(b^2-3*b*c+c^2)*(b+c)*a-b*c*(b+c)^2) : :

X(14286) lies on these lines: {80,900}, {320,350}, {812,4643}, {2827,14288}, {4675,4728}

X(14287) =  TRIAXIAL POINT OF THESE TRIANGLES: ATIK, FUHRMANN, OUTER-GARCIA

Barycentrics    (2*a^4+(b+c)*a^3+(3*b-c)*(b-3*c)*a^2+(b+c)*(3*b^2+2*b*c+3*c^2)*a-(b+c)^4)*(b-c) : :

X(14287) lies on these lines: {11,244}, {2827,14285}

X(14288) =  TRIAXIAL POINT OF THESE TRIANGLES: 2nd CONWAY, FUHRMANN, OUTER-GARCIA

Barycentrics    ((b+c)*a^3+(b^2+c^2)*a^2-b*c*(b+c)^2)*(b-c) : :
X(14288) = X(1769)-3*X(4728)

X(14288) lies on these lines: {11,244}, {513,1577}, {834,2517}, {2827,14286}, {3716,4491}, {3738,13272}, {4036,6371}, {4057,8062}, {4145,4768}, {4391,9002}

X(14288) = reflection of X(i) in X(j) for these (i,j): (4057, 8062), (4491, 3716)

X(14289) =  TRIAXIAL POINT OF THESE TRIANGLES: ABC, 4th ANTI-BROCARD, 1st PARRY

Barycentrics    a^2*(b^2-c^2)*((a^2+b^2+c^2)^2-36*b^2*c^2)*(5*a^2-b^2-c^2) : :

X(14289) lies on these lines: {512,5107}, {1499,4786}

X(14290) =  TRIAXIAL POINT OF THESE TRIANGLES: ABC-X3 REFLECTIONS, KOSNITA, 1st PARRY

Barycentrics    (SB^2-SC^2)*(S^2*(117*R^2-29*SW)+12*S^2*SA-3*SA^2*(9*R^2-SW)) : :

X(14290) lies on these lines: {526,12041}, {1510,9126}

X(14291) =  TRIAXIAL POINT OF THESE TRIANGLES: ABC, 6th ANTI-MIXTILINEAR, 4th EXTOUCH

Barycentrics    (b-c)*(a^2+2*(b+c)*a-b^2-c^2)*(3*a^2-b^2-c^2) : :

X(14291) lies on these lines: {241,514}, {3566,3798}

X(14292) =  TRIAXIAL POINT OF THESE TRIANGLES: ABC, MANDART-EXCIRCLES, 3rd MIXTILINEAR

Barycentrics    a*(b-c)*(a^4-2*(b-c)^2*a^2+(b^2-c^2)^2)*(a^5+(b+c)*a^4-2*(b^2+c^2)*a^3-2*(b+c)*(b^2+c^2)*a^2+(b^4+6*b^2*c^2+c^4)*a+(b^2-c^2)^2*(b+c)) : :

X(14292) lies on the line {513,663}

X(14293) =  TRIAXIAL POINT OF THESE TRIANGLES: ABC-X3 REFLECTIONS, 1st ANTI-CIRCUMPERP, 6th BROCARD

Barycentrics    a^2*((b^2+c^2)*a^6-(b^2+c^2)^2*a^4-(b^4-c^4)*(b^2-c^2)*a^2+b^8+6*b^4*c^4+c^8)*(b^2-c^2)*(a^2-b*c)*(a^2+b*c) : :

X(14293) lies on these lines: {523,2071}, {804,5976}

X(14294) =  TRIAXIAL POINT OF THESE TRIANGLES: ABC-X3 REFLECTIONS, 1st ANTI-CIRCUMPERP, CONWAY

Barycentrics    (b-c)*(a^7-(b^2-b*c+c^2)*a^5+b*c*(b+c)*a^4-(b^4+c^4+2*b*c*(b^2-b*c+c^2))*a^3-2*b*c*(b+c)*(b^2+c^2)*a^2+(b+c)*(b^2-c^2)*(b^3-c^3)*a+(b^2-c^2)^2*(b+c)*b*c) : :

X(14294) lies on these lines: {320,350}, {522,4091}, {523,2071}, {2849,4815}, {4509,8768}

X(14295) =  TRIAXIAL POINT OF THESE TRIANGLES: 1st ANTI-CIRCUMPERP, ANTICOMPLEMENTARY, 6th BROCARD

Barycentrics    (a^4-b^2*c^2)*(b^2-c^2)*b^2*c^2 : :
X(14295) = 2*X(2513)-3*X(5996)

X(14295) lies on these lines: {2,2491}, {76,690}, {290,879}, {325,523}, {338,3124}, {670,888}, {778,9491}, {804,5976}, {1916,2799}, {1926,4374}, {2793,9772}, {7638,7763}

X(14295) = isotomic conjugate of X(805)
X(14295) = anticomplement of X(2491)

X(14296) =  TRIAXIAL POINT OF THESE TRIANGLES: 1st ANTI-CIRCUMPERP, 6th BROCARD, CONWAY

Barycentrics    (b-c)*(a^4-b^2*c^2)*b*c : :

X(14296) lies on these lines: {75,9508}, {76,2787}, {320,350}, {667,7255}, {804,5976}, {1909,4922}, {3261,4874}, {3716,3766}, {4369,4374}

X(14297) =  TRIAXIAL POINT OF THESE TRIANGLES: ANTICOMPLEMENTARY, 6th BROCARD, CONWAY

Barycentrics    (b-c)*(b*c*a^7-b*c*(b+c)*a^6+(b^2+c^2)*(b^2+b*c+c^2)*a^5+(b^3+c^3)*(b^2+c^2)*a^4-(b^6+b^3*c^3+c^6)*a^3-(b+c)*(b^6-b^3*c^3+c^6)*a^2-(b^3+c^3)*(b+c)*b^2*c^2*a-(b^3-c^3)*b^2*c^2*(b^2-c^2))/a : :

X(14297) lies on these lines: {522,693}, {1916,2799}

X(14298) =  TRIAXIAL POINT OF THESE TRIANGLES: EXCENTRAL, EXTANGENTS, EXTOUCH

Barycentrics    a*(b-c)*(a^3+(b+c)*a^2-(b+c)^2*a-(b^2-c^2)*(b-c))*(-a+b+c) : :
X(14298) = 3*X(650)-X(13401) = 2*X(13401)-3*X(14300)

X(14298) lies on these lines: {2,4131}, {6,2431}, {9,2432}, {44,513}, {520,6587}, {521,3239}, {651,7339}, {1364,13609}, {2520,4524}, {3064,3700}, {3309,14330}, {3738,4521}, {4130,8611}, {6129,10397}

X(14298) = midpoint of X(2520) and X(4524)
X(14298) = reflection of X(14300) in X(650)
X(14298) = complement of X(4131)
X(14298) = crossdifference of every pair of points on line X(1)X(84)
X(14298) = isogonal conjugate of X(37141)
X(14298) = crosspoint of X(i) and X(j) for these {i,j}: {1, 37141}, {8, 651}, {9, 1783}, {57, 30239}, {108, 196}, {8058, 14837}
X(14298) = crosssum of X(i) and X(j) for these {i,j}: {1, 14298}, {6, 23224}, {9, 30201}, {56, 650}, {57, 905}, {268, 521}, {513, 3554}, {8059, 36049}
X(14298) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (650, 654, 4394), (657, 661, 650)

X(14299) =  TRIAXIAL POINT OF THESE TRIANGLES: EXCENTRAL, EXTANGENTS, INNER-GARCIA

Barycentrics    a*(b-c)*(a^3-(b^2+c^2)*a-b*c*(b+c))*((b+c)*a^2-2*b*c*a-(b^2-c^2)*(b-c)) : :

X(14299) lies on these lines: {44,513}, {80,3738}, {900,14307}

X(14300) =  TRIAXIAL POINT OF THESE TRIANGLES: EXCENTRAL, EXTANGENTS, HUTSON EXTOUCH

Barycentrics    a*(b-c)*(-a+b+c)*(a^3+(b+c)*a^2-(b^2-6*b*c+c^2)*a-(b^2-c^2)*(b-c)) : :
X(14300) = 2*X(13401)+X(14298)

X(14300) lies on these lines: {44,513}, {521,4765}, {2488,6182}, {3900,4976}, {14282,14303}, {14309,14311}

X(14300) = midpoint of X(650) and X(13401)
X(14300) = reflection of X(14298) in X(650)
X(14300) = {X(650), X(4790)}-harmonic conjugate of X(652)

X(14301) =  TRIAXIAL POINT OF THESE TRIANGLES: EXCENTRAL, EXTOUCH, INNER-GARCIA

Barycentrics    a*(b-c)*(a^8-2*(b+c)*a^7-(2*b^2-7*b*c+2*c^2)*a^6+2*(b+c)*(3*b^2-5*b*c+3*c^2)*a^5-b*c*(11*b^2-10*b*c+11*c^2)*a^4-2*(b+c)*(3*b^4+3*c^4-2*b*c*(5*b^2-6*b*c+5*c^2))*a^3+(2*b^4+2*c^4+b*c*(5*b^2-2*b*c+5*c^2))*(b-c)^2*a^2+2*(b^4-c^4)*(b-c)*(b^2-3*b*c+c^2)*a-(b^2-c^2)^2*(b^4+c^4-3*b*c*(b^2+c^2)))*(-a+b+c) : :

X(14301) lies on these lines: {80,3738}, {521,3239}, {14307,14312}

X(14302) =  TRIAXIAL POINT OF THESE TRIANGLES: EXCENTRAL, EXTOUCH, OUTER-GARCIA

Barycentrics    (b-c)*(-a+b+c)*(a^6-2*(b+c)*a^5-(b+c)^2*a^4+4*(b^3+c^3)*a^3-(b^2-c^2)^2*a^2-2*(b^4-c^4)*(b-c)*a+(b^2-c^2)^2*(b+c)^2) : :

X(14302) lies on these lines: {240,522}, {521,3239}, {4163,8058}

X(14303) =  TRIAXIAL POINT OF THESE TRIANGLES: EXCENTRAL, EXTOUCH, HUTSON EXTOUCH

Barycentrics    a*(b-c)*(a^6-(3*b^2+4*b*c+3*c^2)*a^4-4*b*c*(b+c)*a^3+(3*b^4+3*c^4+2*b*c*(b+2*c)*(2*b+c))*a^2+4*(b^2-c^2)*(b-c)*b*c*a-(b^4-c^4)*(b^2-c^2))*(-a+b+c) : :

X(14303) lies on these lines: {521,3239}, {650,1734}, {14282,14300}, {14309,14313}

X(14304) =  TRIAXIAL POINT OF THESE TRIANGLES: EXCENTRAL, INNER-GARCIA, OUTER-GARCIA

Barycentrics    (2*a^4-(b+c)*a^3-(b-c)^2*a^2+(b^2-c^2)*(b-c)*a-(b^2-c^2)^2)*(b-c)*(-a+b+c)/a : :

X(14304) lies on these lines: {4,2849}, {11,123}, {80,3738}, {240,522}, {521,4086}, {4397,6332}

X(14304) = isogonal conjugate of X(36040)
X(14304) = crossdifference of every pair of points on line X(48)X(1415)

X(14305) =  TRIAXIAL POINT OF THESE TRIANGLES: EXCENTRAL, INNER-GARCIA, HUTSON EXTOUCH

Barycentrics    a*(b-c)*(a^9+5*(b+c)*a^8-(8*b^2+13*b*c+8*c^2)*a^7-(b+c)*(16*b^2+17*b*c+16*c^2)*a^6+(18*b^4+18*c^4+b*c*(19*b^2+58*b*c+19*c^2))*a^5+3*(b+c)*(6*b^4+6*c^4+b*c*(13*b^2-6*b*c+13*c^2))*a^4-(16*b^6+16*c^6-b*c*(b^2-6*b*c+c^2)^2)*a^3-(b+c)*(8*b^6+8*c^6+(27*b^4+27*c^4-2*b*c*(18*b^2-49*b*c+18*c^2))*b*c)*a^2+(b^2-c^2)^2*(5*b^4+5*c^4-7*b*c*(b^2+4*b*c+c^2))*a+(b^2-c^2)^2*(b+c)*(b^4+c^4+5*b*c*(b^2+c^2))) : :

X(14305) lies on these lines: {80,3738}, {14282,14300}

X(14306) =  TRIAXIAL POINT OF THESE TRIANGLES: EXCENTRAL, OUTER-GARCIA, HUTSON EXTOUCH

Barycentrics    (b-c)*(a^7+5*(b+c)*a^6-(7*b^2+6*b*c+7*c^2)*a^5-(b+c)*(11*b^2+34*b*c+11*c^2)*a^4+(11*b^2-10*b*c+11*c^2)*(b+c)^2*a^3+(b+c)*(7*b^4+7*c^4+2*b*c*(18*b^2+5*b*c+18*c^2))*a^2-(b^2-c^2)^2*(5*b^2+6*b*c+5*c^2)*a-(b^2-c^2)^2*(b+c)^3) : :

X(14306) lies on these lines: {240,522}, {14282,14300}, {14311,14313}

X(14307) =  TRIAXIAL POINT OF THESE TRIANGLES: EXTANGENTS, EXTOUCH, INNER-GARCIA

Barycentrics    a*(b-c)*(2*(b+c)*a^6-2*(b^2+3*b*c+c^2)*a^5-(b+c)*(4*b^2-11*b*c+4*c^2)*a^4+4*(b^4+c^4+b*c*(b-c)^2)*a^3+2*(b^3+c^3)*(b^2-4*b*c+c^2)*a^2-2*(b^2-c^2)*(b-c)*(b^3+c^3)*a-(b^2-c^2)^2*(b+c)*b*c)*(-a+b+c) : :

X(14307) lies on these lines: {900,14299}, {3064,3700}, {14301,14312}

X(14308) =  TRIAXIAL POINT OF THESE TRIANGLES: EXTANGENTS, EXTOUCH, OUTER-GARCIA

Barycentrics    (3*a^4-2*(b^2+c^2)*a^2-(b^2-c^2)^2)*(-a+b+c)*(b^2-c^2) : :

X(14308) lies on these lines: {523,656}, {3064,3700}, {4163,8058}

X(14309) =  TRIAXIAL POINT OF THESE TRIANGLES: EXTANGENTS, EXTOUCH, HUTSON EXTOUCH

Barycentrics    a*(b-c)*(a^9-(b+c)*a^8-4*(b^2+c^2)*a^7+4*(b+c)*(b^2+c^2)*a^6+6*(b^2-c^2)^2*a^5-2*(b+c)*(3*b^2+c^2)*(b^2+3*c^2)*a^4-4*(b^4+c^4-2*b*c*(b^2+b*c+c^2))*(b+c)^2*a^3+4*(b^2-c^2)*(b-c)*(b^4+c^4+2*b*c*(b^2+3*b*c+c^2))*a^2+(b^2-c^2)^4*a-(b^2-c^2)^4*(b+c))*(-a+b+c) : :

X(14309) lies on these lines: {3064,3700}, {14300,14311}, {14303,14313}

X(14310) =  TRIAXIAL POINT OF THESE TRIANGLES: EXTANGENTS, INNER-GARCIA, OUTER-GARCIA

Barycentrics    ((b+c)*a^4+(b-c)^2*a^3-(b^3+c^3)*a^2-(b-c)^2*(b^2+c^2)*a-(b^2-c^2)*(b-c)*b*c)*(b^2-c^2)*(-a+b+c) : :

X(14310) lies on these lines: {11,123}, {523,656}, {900,14299}, {2803,6284}, {2850,7354}, {3028,3324}, {6044,6089}

X(14311) =  TRIAXIAL POINT OF THESE TRIANGLES: EXTANGENTS, OUTER-GARCIA, HUTSON EXTOUCH

Barycentrics    (a^9-5*(b+c)*a^8+16*(b+c)*(b^2+c^2)*a^6-2*(3*b^4+34*b^2*c^2+3*c^4)*a^5-6*(b+c)*(3*b^4+2*b^2*c^2+3*c^4)*a^4+8*(b^2+c^2)*(b^4+6*b^2*c^2+c^4)*a^3+8*(b^4-c^4)*(b^2-c^2)*(b+c)*a^2-3*(b^2-c^2)^4*a-(b^2-c^2)^4*(b+c))*(b^2-c^2) : :

X(14311) lies on these lines: {523,656}, {14300,14309}, {14306,14313}

X(14312) =  TRIAXIAL POINT OF THESE TRIANGLES: EXTOUCH, INNER-GARCIA, OUTER-GARCIA

Barycentrics    ((b+c)*a^5-(b^2+c^2)*a^4-2*(b^2-c^2)*(b-c)*a^3+2*(b^2-c^2)^2*a^2+(b^2-c^2)*(b-c)^3*a-(b^4-c^4)*(b^2-c^2))*(b-c)*(-a+b+c) : :

X(14312) lies on these lines: {4,6087}, {11,123}, {104,900}, {403,523}, {522,905}, {4163,8058}, {14301,14307}

X(14313) =  TRIAXIAL POINT OF THESE TRIANGLES: EXTOUCH, OUTER-GARCIA, HUTSON EXTOUCH

Barycentrics    (b-c)*(a^12+2*(b+c)*a^11-4*(b^2+c^2)*a^10-2*(b+c)*(5*b^2+12*b*c+5*c^2)*a^9+5*(b^2-c^2)^2*a^8+4*(b+c)*(5*b^4+5*c^4+2*b*c*(8*b^2+7*b*c+8*c^2))*a^7+16*b^2*c^2*(b+c)^2*a^6-4*(b+c)*(5*b^6+5*c^6+(12*b^4+12*c^4+b*c*(19*b^2-8*b*c+19*c^2))*b*c)*a^5-5*(b^2-c^2)^4*a^4+2*(b^2-c^2)*(b-c)*(5*b^6+5*c^6+(10*b^4+10*c^4+3*b*c*(9*b^2+4*b*c+9*c^2))*b*c)*a^3+4*(b^2-c^2)^2*(b+c)^2*(b^4+c^4-2*b*c*(b^2+b*c+c^2))*a^2-2*(b^2-c^2)^4*(b+c)*(b^2-4*b*c+c^2)*a-(b^2-c^2)^6)*(-a+b+c) : :

X(14313) lies on these lines: {4163,8058}, {14303,14309}, {14306,14311}

X(14314) =  TRIAXIAL POINT OF THESE TRIANGLES: ABC, ANTI-ORTHOCENTROIDAL, TRINH

Barycentrics    a^2*(b^2-c^2)*(-a^2+b^2+c^2)*(a^4+(b^2+c^2)*a^2-2*(b^2-c^2)^2)*(a^2-b^2-b*c-c^2)*(a^2-b^2+b*c-c^2) : :

X(14314) lies on these lines: {265,6334}, {340,3268}, {520,647}, {526,1511}

X(14315) =  TRIAXIAL POINT OF THESE TRIANGLES: 2nd CIRCUMPERP, OUTER-GARCIA, 2nd SHARYGIN

Barycentrics    a*(b-c)*(2*(b+c)*a^2-2*b*c*a-(b+c)*(2*b^2-3*b*c+2*c^2)) : :
X(14315) = 3*X(1769)+X(2254)

X(14315) lies on these lines: {11,244}, {513,1960}, {523,10015}, {918,4364}, {4017,4977}, {4389,4453}

X(14316) =  TRIAXIAL POINT OF THESE TRIANGLES: 1st ANTI-BROCARD, ANTICOMPLEMENTARY, 2nd NEUBERG

Barycentrics    (a^6+b^2*c^2*a^2-b^6-c^6)*(b^2-c^2) : :
X(14316) = 3*X(1637)-X(6333) = X(3569)-3*X(9979)

X(14316) lies on these lines: {297,525}, {339,3124}, {523,5027}, {690,13187}, {826,4142}, {1112,9517}, {1637,6333}, {12188,13519}

X(14317) =  TRIAXIAL POINT OF THESE TRIANGLES: ANTICOMPLEMENTARY, 3rd BROCARD, SYMMEDIAL

Barycentrics    a^2*((b^2-c^2)^2*a^8+2*(b^6+c^6)*a^6-(b^4+b^2*c^2+c^4)*b^2*c^2*a^4-b^6*c^6)*(b^2-c^2) : :

X(14317) lies on these lines: {669,2451}, {694,5027}

X(14318) =  TRIAXIAL POINT OF THESE TRIANGLES: ABC, ARTZT, 2nd NEUBERG

Barycentrics    a^2*(b^2-c^2)*(a^4+2*(b^2+c^2)*a^2+b^2*c^2) : :
X(14318) = 3*X(669)-X(3288) = 2*X(3288)-3*X(5027)

X(14318) lies on these lines: {6,688}, {187,237}, {690,5987}, {5652,5888}, {9147,12073}

X(14318) = reflection of X(i) in X(j) for these (i,j): (3005, 5113), (5027, 669)

X(14319) =  TRIAXIAL POINT OF THESE TRIANGLES: 4th EULER, EXCENTERS-MIDPOINTS, FEUERBACH

Barycentrics    (a^6-3*(b+c)*a^5-(b^2-3*b*c+c^2)*a^4+2*(b+c)*(3*b^2+b*c+3*c^2)*a^3-((b^2-c^2)^2-4*b^2*c^2)*a^2-(b+c)*(b^2+c^2)*(3*b^2-2*b*c+3*c^2)*a+(b+c)*(b^2-c^2)*(b^3-c^3))*(b^2-c^2) : :

X(14319) lies on these lines: {9,14321}, {523,1577}

X(14320) =  TRIAXIAL POINT OF THESE TRIANGLES: 4th EULER, FEUERBACH, HUTSON EXTOUCH

Barycentrics    (2*a^9+5*(b+c)*a^8-(11*b^2+16*b*c+11*c^2)*a^7-(b+c)*(17*b^2+39*b*c+17*c^2)*a^6+(21*b^4+21*c^4+2*b*c*(12*b^2+13*b*c+12*c^2))*a^5+(b+c)*(21*b^4+21*c^4+b*c*(77*b^2+96*b*c+77*c^2))*a^4-(17*b^6+17*c^6-3*b^2*c^2*(35*b^2+64*b*c+35*c^2))*a^3-(b^2-c^2)*(b-c)*(11*b^4+11*c^4+b*c*(59*b^2+84*b*c+59*c^2))*a^2+(b^2-c^2)^2*(5*b^4+5*c^4-2*b*c*(4*b^2+7*b*c+4*c^2))*a+(b^2-c^2)^3*(b-c)*(2*b^2+3*b*c+2*c^2))*(b^2-c^2) : :

X(14320) lies on these lines: {523,1577}, {14323,14324}

X(14321) =  TRIAXIAL POINT OF THESE TRIANGLES: EXCENTERS-MIDPOINTS, 2nd EXTOUCH, FEUERBACH

Barycentrics    (3*a-b-c)*(b^2-c^2) : :
X(14321) = X(649)-3*X(1639) = 3*X(661)+X(4024) = X(661)+3*X(4120) = 7*X(661)+X(4838) = 3*X(661)-X(4841) = 5*X(661)+3*X(4931) = 5*X(661)-X(4988) = 3*X(1639)-2*X(2490) = 2*X(2529)-7*X(3239) = 3*X(3700)-X(4024)

X(14321) lies on these lines: {2,2487}, {9,14319}, {10,1499}, {37,7180}, {101,2692}, {513,2529}, {514,4940}, {523,661}, {525,4129}, {649,1639}, {650,900}, {918,3776}, {1826,2501}, {2527,4790}, {2976,3667}, {3004,4776}, {4106,4468}, {4705,4843}, {4765,4926}, {4893,4976}, {4944,4977}, {6008,11068}

X(14321) = midpoint of X(i) and X(j) for these {i,j}: {661, 3700}, {4024, 4841}, {4106, 4468}, {4394, 4949}
X(14321) = reflection of X(i) in X(j) for these (i,j): (649, 2490), (4394, 4521), (4790, 2527), (4897, 2487)
X(14321) = complement of X(4897)
X(14321) = anticomplement of X(2487)
X(14321) = pole wrt Spieker circle of line X(2)X(6)
X(14321) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 4897, 2487), (649, 1639, 2490), (661, 4024, 4841), (661, 4120, 3700), (661, 4931, 4988), (3700, 4841, 4024)

X(14322) =  TRIAXIAL POINT OF THESE TRIANGLES: EXCENTERS-MIDPOINTS, 2nd EXTOUCH, HUTSON EXTOUCH

Barycentrics    (b-c)*(3*a^8-6*(b+c)*a^7-(4*b^2+b*c+4*c^2)*a^6+7*(b+c)*(2*b^2-b*c+2*c^2)*a^5-2*(b^4+c^4-b*c*(5*b^2+14*b*c+5*c^2))*a^4-2*(b+c)*(5*b^4+5*c^4-b*c*(3*b^2+2*b*c+3*c^2))*a^3+(4*b^4+4*c^4-b*c*(17*b^2-18*b*c+17*c^2))*(b+c)^2*a^2+(b^2-c^2)^2*(b+c)*(2*b^2+b*c+2*c^2)*a-(b^2-c^2)^4) : :

X(14322) lies on these lines: {}

X(14323) =  TRIAXIAL POINT OF THESE TRIANGLES: EXCENTERS-MIDPOINTS, FEUERBACH, HUTSON EXTOUCH

Barycentrics    (b-c)*(12*a^8-21*(b+c)*a^7-(19*b^2+10*b*c+19*c^2)*a^6+(b+c)*(47*b^2-6*b*c+47*c^2)*a^5+(b^4+c^4+2*b*c*(10*b^2+27*b*c+10*c^2))*a^4-(b+c)*(31*b^4+31*c^4-2*b*c*(6*b^2-5*b*c+6*c^2))*a^3+(b^2-c^2)^2*(7*b^2-10*b*c+7*c^2)*a^2+(b^2-c^2)^2*(b+c)*(5*b^2-6*b*c+5*c^2)*a-(b^2-c^2)^4) : :

X(14323) lies on these lines: {9,14319}, {14320,14324}

X(14324) =  TRIAXIAL POINT OF THESE TRIANGLES: 2nd EXTOUCH, FEUERBACH, HUTSON EXTOUCH

Barycentrics    (3*a^3-5*(b+c)*a^2+(b-c)^2*a+(b^2-c^2)*(b-c))*(b^2-c^2) : :

X(14324) lies on these lines: {514,4130}, {523,661}, {3709,7178}, {14300,14309}, {14320,14323}

X(14325) =  TRIAXIAL POINT OF THESE TRIANGLES: ABC, INNER-SQUARES, OUTER-VECTEN

Barycentrics    (b^2-c^2)*(a^2+b^2+c^2+4*S)*(-a^2+b^2+c^2+2*S)*(a^2+S) : :

X(14325) lies on these lines: {230,231}, {9131,13320}

X(14326) =  TRIAXIAL POINT OF THESE TRIANGLES: ABC, OUTER-SQUARES, INNER-VECTEN

Barycentrics    (b^2-c^2)*(a^2+b^2+c^2-4*S)*(-a^2+b^2+c^2-2*S)*(a^2-S) : :

X(14326) lies on these lines: {230,231}, {9131,13319}

X(14327) =  TRIAXIAL POINT OF THESE TRIANGLES: ABC, ANTI-ARTZT, LEMOINE

Barycentrics    (4*a^4-(b^2+c^2)*a^2-4*b^2*c^2+c^4+b^4)*(a^4+5*(b^2+c^2)*a^2+2*b^2*c^2-2*c^4-2*b^4)*(b^2-c^2) : :

X(14327) lies on these lines: {351,523}, {2793,9135}, {5640,6088}

X(14328) =  TRIAXIAL POINT OF THESE TRIANGLES: 4th ANTI-BROCARD, CIRCUMSYMMEDIAL, 1st EHRMANN

Barycentrics    a^2*(13*a^10-34*(b^2+c^2)*a^8-(74*b^4-263*b^2*c^2+74*c^4)*a^6+4*(b^2+c^2)*(b^2+6*b*c+c^2)*(b^2-6*b*c+c^2)*a^4+(29*b^8+29*c^8-b^2*c^2*(109*b^4-372*b^2*c^2+109*c^4))*a^2-2*(b^6+c^6)*(b^2+c^2)^2)*(b^2-c^2) : :

X(14328) lies on these lines: {}

X(14329) =  TRIAXIAL POINT OF THESE TRIANGLES: ABC-X3 REFLECTIONS, ARIES, TANGENTIAL

Barycentrics    a^2*(2*a^8-5*(b^2+c^2)*a^6+(3*b^4+8*b^2*c^2+3*c^4)*a^4+((b^2-c^2)^2-4*b^2*c^2)*(b^2+c^2)*a^2-(b^4+c^4)*(b^2-c^2)^2)*(b^2-c^2)*(-a^2+b^2+c^2) : :

X(14329) lies on these lines: {520,11589}, {669,684}, {8057,9409}

X(14330) =  TRIAXIAL POINT OF THESE TRIANGLES: ABC, ATIK, 7th MIXTILINEAR

Barycentrics    (b-c)*(3*a^2+(b-c)^2)*(-a+b+c)^2 : :

X(14330) lies on these lines: {9,522}, {241,514}, {523,14282}, {1108,6586}, {2291,2724}, {3126,8568}, {3239,3900}, {3309,14298}, {4147,4521}, {4765,6362}, {6366,10006}

X(14330) = {X(3239), X(4148)}-harmonic conjugate of X(4163)

X(14331) =  TRIAXIAL POINT OF THESE TRIANGLES: EXTOUCH, 3rd EXTOUCH, 5th EXTOUCH

Barycentrics    (b-c)*(3*a^4-2*(b^2+c^2)*a^2-(b^2-c^2)^2)*(-a+b+c) : :

X(14331) lies on these lines: {1,10397}, {241,514}, {243,522}, {521,3239}, {657,6590}, {6587,8057}, {6591,7661}

X(14331) = midpoint of X(652) and X(3064)

X(14332) =  TRIAXIAL POINT OF THESE TRIANGLES: ABC, ANTI-CONWAY, 2nd BROCARD

Barycentrics    (SB^2-SC^2)*(3*SA-SW)*(4*SW*(2*R^2-SW)+3*S^2+3*SA^2) : :

X(14332) lies on these lines: {351,690}, {512,2623}

X(14333) =  TRIAXIAL POINT OF THESE TRIANGLES: MEDIAL, MIDHEIGHT, INNER-SQUARES

Barycentrics    (2*(-a^2+b^2+c^2)*S+3*a^4-2*(b^2+c^2)*a^2-(b^2-c^2)^2)*(b^2-c^2) : :

X(14333) lies on these lines: {485,5466}, {523,14335}, {525,3239}, {9131,13319}

X(14333) = reflection of X(14334) in X(6587)

X(14334) =  TRIAXIAL POINT OF THESE TRIANGLES: MEDIAL, MIDHEIGHT, OUTER-SQUARES

Barycentrics    (-2*(-a^2+b^2+c^2)*S+3*a^4-2*(b^2+c^2)*a^2-(b^2-c^2)^2)*(b^2-c^2) : :

X(14334) lies on these lines: {486,5466}, {523,14336}, {525,3239}, {9131,13320}

X(14334) = reflection of X(14333) in X(6587)

X(14335) =  TRIAXIAL POINT OF THESE TRIANGLES: MIDHEIGHT, INNER-SQUARES, SUBMEDIAL

Barycentrics    (SB-SC)*((2*SA-SW)^2+2*(SA-SW)*S+2*S^2) : :

X(14335) lies on these lines: {512,14336}, {523,14333}

X(14336) =  TRIAXIAL POINT OF THESE TRIANGLES: MIDHEIGHT, OUTER-SQUARES, SUBMEDIAL

Barycentrics    (SB-SC)*((2*SA-SW)^2-2*(SA-SW)*S+2*S^2) : :

X(14336) lies on these lines: {512,14335}, {523,14334}

X(14337) =  TRIAXIAL POINT OF THESE TRIANGLES: 2nd BROCARD, CIRCUMSYMMEDIAL, INNER-GREBE

Barycentrics    a^2*(2*(a^2+b^2+c^2)*(2*a^2-b^2-c^2)*S-2*a^6-12*b^2*c^2*a^2-(b^2+c^2)*(2*b^4-11*b^2*c^2+2*c^4))*(b^2-c^2) : :

X(14337) lies on these lines: {351,14338}, {690,14339}

X(14337) = reflection of X(14338) in X(351)

X(14338) =  TRIAXIAL POINT OF THESE TRIANGLES: 2nd BROCARD, CIRCUMSYMMEDIAL, OUTER-GREBE

Barycentrics    a^2*(2*(a^2+b^2+c^2)*(2*a^2-b^2-c^2)*S+2*a^6+12*b^2*c^2*a^2+(b^2+c^2)*(2*b^4-11*b^2*c^2+2*c^4))*(b^2-c^2) : :

X(14338) lies on these lines: {351,14337}, {690,14340}

X(14338) = reflection of X(14337) in X(351)

X(14339) =  TRIAXIAL POINT OF THESE TRIANGLES: 2nd BROCARD, INNER-GREBE, TANGENTIAL

Barycentrics    a^2*((a^2+b^2+c^2)*(2*a^2-b^2-c^2)*S-a^6-3*b^2*c^2*a^2-(b^2+c^2)*(b^4-4*b^2*c^2+c^4))*(b^2-c^2) : :

X(14339) lies on these lines: {351,3455}, {512,6567}, {690,14337}

X(14339) = reflection of X(14340) in X(351)

X(14340) =  TRIAXIAL POINT OF THESE TRIANGLES: 2nd BROCARD, OUTER-GREBE, TANGENTIAL

Barycentrics    a^2*((a^2+b^2+c^2)*(2*a^2-b^2-c^2)*S+a^6+3*b^2*c^2*a^2+(b^2+c^2)*(b^4-4*b^2*c^2+c^4))*(b^2-c^2) : :

X(14340) lies on these lines: {351,3455}, {512,6566}, {690,14338}

X(14340) = reflection of X(14339) in X(351)

X(14341) =  TRIAXIAL POINT OF THESE TRIANGLES: MEDIAL, MIDHEIGHT, SUBMEDIAL

Barycentrics    (5*a^4-4*(b^2+c^2)*a^2-(b^2-c^2)^2+4*b^2*c^2)*(b^2-c^2) : :
X(14341) = 3*X(2)+X(2501) = 9*X(2)-X(6563) = X(850)+3*X(9209) = 3*X(1637)+X(3265) = 3*X(2501)+X(6563) = X(6562)+3*X(9134)

X(14341) lies on these lines: {2,2501}, {5,1499}, {512,14335}, {525,3239}, {669,5020}, {850,9209}, {1637,3265}, {5926,6642}, {6130,8057}, {6562,9134}, {6677,10189}, {9009,9822}

X(14342) =  TRIAXIAL POINT OF THESE TRIANGLES: ABC, ANTI-EXCENTERS-REFLECTIONS, 2nd EXTOUCH

Barycentrics    -(b-c)*(-a+b+c)*(a^6-(b+c)*a^5+2*(b+c)*(b^2+c^2)*a^3-3*(b^2-c^2)^2*a^2-(b^2-c^2)^2*(b+c)*a+2*(b^2-c^2)*(b^4-c^4)) : :

X(14342) lies on these lines: {522,650}, {523,10151}, {3667,14344}, {3900,4064}, {7178,14302}

X(14342) = reflection of X(7178) in X(14302)

X(14343) =  TRIAXIAL POINT OF THESE TRIANGLES: ABC, ANTI-INVERSE-IN-INCIRCLE, MIDHEIGHT

Barycentrics    (b^2-c^2)*(a^4+2*(b^2+c^2)*a^2-2*b^2*c^2-3*c^4-3*b^4)*(3*a^4-2*(b^2+c^2)*a^2-(b^2-c^2)^2) : :

X(14343) lies on these lines: {325,523}, {6587,8057}

X(14343) = isotomic conjugate of X(35571)

X(14344) =  TRIAXIAL POINT OF THESE TRIANGLES: ABC, CIRCUMORTHIC, 3rd EXTOUCH

Barycentrics    (b-c)*(2*a^6-(b+c)*a^5-3*(b^2+c^2)*a^4+2*(b+c)*(b^2+c^2)*a^3-(b^2-c^2)^2*(b+c)*a+(b^4-c^4)*(b^2-c^2)) : :

X(14344) lies on these lines: {186,523}, {241,514}, {440,1577}, {918,4091}, {1817,4560}, {3651,6362}, {3667,14342}

X(14345) =  TRIAXIAL POINT OF THESE TRIANGLES: ABC, MIDHEIGHT, ORTHOCENTROIDAL

Barycentrics    (b^2-c^2)*(-a^2+b^2+c^2)*(3*a^4-2*(b^2+c^2)*a^2-(b^2-c^2)^2)*(2*a^4-(b^2+c^2)*a^2-(b^2-c^2)^2) : :

X(14345) lies on these lines: {520,9209}, {647,6368}, {1636,1637}, {5972,13526}, {6587,8057}

X(14345) = midpoint of X(1636) and X(1637)
X(14345) = tripolar centroid of X(20)

X(14346) =  TRIAXIAL POINT OF THESE TRIANGLES: MIDHEIGHT, ORTHIC, REFLECTION

Barycentrics    a^2*(b^2-c^2)*(a^8-4*(b^2+c^2)*a^6+(6*b^4+5*b^2*c^2+6*c^4)*a^4-2*(b^2+c^2)*(2*b^4-b^2*c^2+2*c^4)*a^2+(b^4+3*b^2*c^2+c^4)*(b^2-c^2)^2) : :

X(14346) lies on these lines: {520,6587}, {526,2501}, {1510,12077}

X(14347) =  TRIAXIAL POINT OF THESE TRIANGLES: ABC, ANDROMEDA, ANTLIA

Barycentrics    a*((a-b)^2+3*c^2)*((a-c)^2+3*b^2)*(b-c)*((b+c)*a-b^2-c^2)*(a^2+(b-c)^2) : :

X(14347) lies on the line {2254,3669}

X(14348) =  TRIAXIAL POINT OF THESE TRIANGLES: ABC, ANTLIA, 2nd CIRCUMPERP

Barycentrics    a*(b-c)*(a^2+(b-c)^2)*(a^4+2*b*c*a^2-4*(b+c)*(b^2+c^2)*a+(b^2+c^2)*(3*b^2+2*b*c+3*c^2)) : :

X(14348) lies on the line {36,238}

X(14349) =  TRIAXIAL POINT OF THESE TRIANGLES: ABC, 2nd CIRCUMPERP, 4th CONWAY

Barycentrics    a*(b-c)*((b+c)*a+b^2+b*c+c^2) : :
X(14349) = 2*X(3960)+X(4813) = 2*X(4129)-3*X(4776) = X(4391)-3*X(4776) = X(4498)-3*X(4893) = 3*X(4728)-2*X(4823) = X(4905)+2*X(4983)

X(14349) lies on these lines: {36,238}, {512,1491}, {514,661}, {522,4170}, {523,4992}, {525,3004}, {650,4063}, {663,830}, {784,4010}, {824,4079}, {891,4490}, {2254,4822}, {2526,3309}, {2786,4502}, {3777,6372}, {3960,4813}, {4083,4705}, {4160,4449}, {4498,4893}, {4834,9508}

X(14349) = midpoint of X(i) and X(j) for these {i,j}: {2254, 4822}, {2530, 4983}
X(14349) = reflection of X(i) in X(j) for these (i,j): (1019, 905), (1577, 3835), (1734, 1491), (4063, 650), (4391, 4129), (4834, 9508), (4905, 2530)
X(14349) = isotomic conjugate of X(37218)
X(14349) = {X(4391), X(4776)}-harmonic conjugate of X(4129)

X(14350) =  TRIAXIAL POINT OF THESE TRIANGLES: ABC, 4th CONWAY, EXCENTERS-MIDPOINTS

Barycentrics    (b-c)*(a-3*b-3*c)*(3*a-b-c) : :
X(14350) = X(661)+3*X(3239) = 5*X(661)+3*X(6590) = 9*X(1639)-X(4790) = 5*X(3239)-X(6590) = X(4394)-3*X(4521) = 5*X(4394)+3*X(4949) = X(4394)+3*X(14321) = 4*X(4394)-3*X(14351) = 5*X(4521)+X(4949) = 4*X(4521)-X(14351) = X(4949)-5*X(14321)

X(14350) lies on these lines: {514,661}, {650,4962}, {1639,4790}, {2490,6006}, {2976,3667}, {4120,4765}

X(14350) = midpoint of X(4521) and X(14321)

X(14351) =  TRIAXIAL POINT OF THESE TRIANGLES: ABC, 5th CONWAY, EXCENTERS-MIDPOINTS

Barycentrics    (b-c)*(5*a+b+c)*(3*a-b-c) : :
X(14351) = 3*X(649)+X(4765) = 15*X(649)+X(4988) = 3*X(4394)-X(4521) = 9*X(4394)-X(4949) = 5*X(4394)-X(14321) = 4*X(4394)-X(14350) = 3*X(4521)-X(4949) = 5*X(4521)-3*X(14321) = 4*X(4521)-3*X(14350) = 5*X(4765)-X(4988) = 5*X(4949)-9*X(14321)

X(14351) lies on these lines: {239,514}, {522,2527}, {2516,6006}, {2976,3667}, {3239,4958}, {3700,4962}

X(14352) =  TRIAXIAL POINT OF THESE TRIANGLES: ABC, EXCENTERS-MIDPOINTS, EXCENTERS-REFLECTIONS

Barycentrics    a*(b-c)*(3*a-b-c)*(a^2+2*(b+c)*a-14*b*c+c^2+b^2) : :

X(14352) lies on these lines: {513,4162}, {2976,3667}

X(14353) =  TRIAXIAL POINT OF THESE TRIANGLES: ANDROMEDA, ANTI-AQUILA, 2nd CIRCUMPERP

Barycentrics    a*(b-c)*(a^3+3*(b+c)*a^2-(b^2+c^2)*a-(b+c)*(3*b^2-4*b*c+3*c^2)) : :

X(14353) lies on these lines: {1,7655}, {513,1960}, {656,14077}, {905,4017}, {2826,7661}, {3309,6129}

X(14353) = midpoint of X(i) and X(j) for these {i,j}: {1, 7655}, {905, 4017}

X(14354) =  ISOGONAL CONJUGATE OF X(1117)

Barycentrics    a^2*(a^2 - b^2 - b*c - c^2)*(a^2 - b^2 + b*c - c^2)*(a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6 - 3*a^4*c^2 - a^2*b^2*c^2 + b^4*c^2 + 3*a^2*c^4 + b^2*c^4 - c^6)*(a^8 + 2*a^6*b^2 - 6*a^4*b^4 + 2*a^2*b^6 + b^8 - 4*a^6*c^2 + a^4*b^2*c^2 + a^2*b^4*c^2 - 4*b^6*c^2 + 6*a^4*c^4 + a^2*b^2*c^4 + 6*b^4*c^4 - 4*a^2*c^6 - 4*b^2*c^6 + c^8)*(a^8 - 4*a^6*b^2 + 6*a^4*b^4 - 4*a^2*b^6 + b^8 + 2*a^6*c^2 + a^4*b^2*c^2 + a^2*b^4*c^2 - 4*b^6*c^2 - 6*a^4*c^4 + a^2*b^2*c^4 + 6*b^4*c^4 + 2*a^2*c^6 - 4*b^2*c^6 + c^8) : :

X(14354) lies on the cubic K073 and this line: {3, 1138}

X(14354) = isogonal conjugate of X(1117)
X(14354) = crosspoint of X(3470) and X(8487)
X(14354) = crosssum of X(3471) and X(5671)
X(14354) = circumcircle-inverse of X(1138)
X(14354) = barycentric quotient X(6)/X(1117)

X(14355) =  X(2)X(98)∩X(3)X(249)

Barycentrics    a^2*(a^2 - b^2 - b*c - c^2)*(a^2 - b^2 + b*c - c^2)*(a^4 + b^4 - a^2*c^2 - b^2*c^2)*(a^4 - a^2*b^2 - b^2*c^2 + c^4) : :

X(14355) lies on the cubics K072, K280, K527, K759 and these lines: on lines {2, 98}, {3, 249}, {4, 6328}, {6, 842}, {54, 826}, {186, 2088}, {248, 574}, {567, 7827}, {575, 13137}, {1138, 7739}, {2065, 5050}, {2909, 9873}, {2966, 7757}, {5028, 11610}, {5039, 9474}, {7799, 10411}

X(14355) = isogonal conjugate of X(14356)
X(14355) = Brocard-circle-inverse of X(11005)
X(14355) = X(i)-isoconjugate of X(j) for these (i,j): {94, 1755}, {240, 265}, {511, 2166}, {1959, 1989}
X(14355) = trilinear pole of line {50, 526}
X(14355) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1976, 11653, 98), (13414, 13415, 11005)
X(14355) = barycentric product X(i)*X(j) for these {i,j}: {50, 290}, {98, 323}, {186, 287}, {248, 340}, {526, 2966}, {685, 8552}, {1821, 6149}, {1976, 7799}, {2395, 10411}, {2715, 3268}
X(14355) = barycentric quotient X(i)/X(j) for these {i,j}: {50, 511}, {98, 94}, {186, 297}, {248, 265}, {287, 328}, {323, 325}, {526, 2799}, {1910, 2166}, {1976, 1989}, {2088, 868}, {2395, 10412}, {2715, 476}, {6149, 1959}, {6531, 6344}, {8552, 6333}, {10411, 2396}, {14270, 3569}

X(14356) =  X(2)X(476)∩X(5)X(523)

Barycentrics    (a^2 - a*b + b^2 - c^2)*(a^2 + a*b + b^2 - c^2)*(a^2 - b^2 - a*c + c^2)*(a^2 - b^2 + a*c + c^2)*(a^2*b^2 - b^4 + a^2*c^2 - c^4) : :
X(14356) = 3 X(3545) + X(9214)

Let ABC be a triangle with anticomplementary triangle A'B'C'. Let DEF and D'E'F' be the cevian triangles of Steiner point and its isotomic conjugate, respectively. Let Ao, Bo, Co be the circumcenters of triangles A'DD', B'EE', C'FF', respectively. Then the triangles ABC and AoBoCo are perspective and the perspector is X(14356). (Angel Montesdeoca, July 18, 2022)

X(14356) lies on the cubics K072, K281, K501, K762, K835 and these lines: {2, 476}, {3, 3447}, {4, 250}, {5, 523}, {6, 13}, {94, 262}, {114, 5968}, {264, 328}, {511, 868}, {827, 1141}, {1485, 5961}, {1995, 3425}, {2065, 5967}, {3545, 5627}, {3580, 13448}, {5055, 13162}

X(14356) = isogonal conjugate of X(14355)
X(14356) = crossdifference of every pair of points on line {50, 526}
X(14356) = X(i)-isoconjugate of X(j) for these (i,j): {50, 1821}, {98, 6149}, {186, 293}, {323, 1910}, {2624, 2966}
X(14356) = X(265)-Hirst inverse of X(1989)
X(14356) = barycentric product X(i)*X(j) for these {i,j}: {94, 511}, {232, 328}, {265, 297}, {325, 1989}, {476, 2799}, {1959, 2166}, {2421, 10412}
X(14356) = barycentric quotient X(i)/X(j) for these {i,j}: {94, 290}, {232, 186}, {237, 50}, {265, 287}, {297, 340}, {325, 7799}, {476, 2966}, {511, 323}, {684, 8552}, {1755, 6149}, {1989, 98}, {2166, 1821}, {2421, 10411}, {2491, 14270}, {2799, 3268}, {3569, 526}, {6530, 14165}, {8430, 9213}, {11060, 1976}

X(14357) =  X(2)X(10415)∩X(3)X(67)

Barycentrics    (2*a^2 - b^2 - c^2)*(a^4 - a^2*b^2 + b^4 - c^4)*(a^4 - b^4 - a^2*c^2 + c^4) : :

X(14357) lies on the cubics K009, K072, K284 and these lines: {2, 10415}, {3, 67}, {4, 842}, {39, 647}, {76, 7664}, {126, 3788}, {1656, 13162}, {3292, 8030}, {5467, 7813}, {6390, 9177}, {7746, 8791}, {9076, 12074}

X(14357) = isogonal conjugate of X(14246)
X(14357) = X(i)-complementary conjugate of X(j) for these (i,j): {922, 1560}, {1177, 4892}
X(14357) = X(i)-Ceva conjugate of X(j) for these (i,j): {935, 690}, {10415, 67}
X(14357) = cevapoint of X(2482) and X(7813)
X(14357) = crosspoint of X(67) and X(10415)
X(14357) = crossdifference of every pair of points on line {23, 2492}
X(14357) = crosssum of X(23) and X(6593)
X(14357) = Cundy-Parry Phi transform of X(542)
X(14357) = Cundy-Parry Psi transform of X(842)
X(14357) = X(i)-isoconjugate of X(j) for these (i,j): {1, 14246}, {23, 897}, {316, 923}, {662, 10561}, {1101, 10555}
X(14357) = barycentric product X(i)*X(j) for these {i,j}: {67, 524}, {2157, 14210}, {2482, 10415}, {3266, 3455}, {6390, 8791}, {7813, 9076}
X(14357) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 14246}, {67, 671}, {115, 10555}, {187, 23}, {351, 2492}, {512, 10561}, {524, 316}, {690, 9979}, {2157, 897}, {2482, 7664}, {3455, 111}

X(14358) =  ISOGONAL CONJUGATE OF X(202)

Barycentrics    Sin(A) / (1 + Sin(A - Pi/6)) : :

X(14358) lies on the cubic K058 and these lines: {1, 3411}, {13, 484}, {519, 5240}, {3638, 3911}

X(14358) = isogonal conjugate of X(202)
X(14358) = X(1)-cross conjugate of X(13)
X(14358) = X(i)-isoconjugate of X(j) for these (i,j): {1, 202}, {15, 3179}
X(14358) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 202}, {2153, 3179}, {7006, 11131}

X(14359) =  ISOGONAL CONJUGATE OF X(203)

Barycentrics    Sin(A) / (1 - Sin(A + Pi/6)) : :

X(14359) lies on the cubic K058 and these lines: {1, 3412}, {14, 484}, {80, 7126}, {519, 5239}, {3639, 3911}

X(14359) = isogonal conjugate of X(203)
X(14359) = X(1)-cross conjugate of X(14)
X(14359) = X(i)-isoconjugate of X(j) for these (i,j): {1, 203}, {5239, 7051}
X(14359) = cevapoint of X(i) and X(j) for these (i,j): {1, 7150}
X(14359) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 203}, {7005, 11130}, {7126, 5239}

X(14360) =  ISOTOMIC CONJUGATE OF X(13574)

Barycentrics    a^6 + a^4*b^2 - a^2*b^4 - b^6 + a^4*c^2 - 5*a^2*b^2*c^2 + 3*b^4*c^2 - a^2*c^4 + 3*b^2*c^4 - c^6 : :
X(14360) = 3 X(2) - 4 X(126) = 5 X(3091) - 4 X(5512) = 9 X(2) - 8 X(6719) = 3 X(111) - 4 X(6719) = 3 X(126) - 2 X(6719) = 10 X(6719) - 9 X(9172) = 5 X(111) - 6 X(9172) = 5 X(2) - 4 X(9172) = 5 X(126) - 3 X(9172) = 4 X(6719) - 9 X(10717), 2 X(9172) - 5 X(10717), X(111) - 3 X(10717), 2 X(126) - 3 X(10717), 5 X(3616) - 4 X(11721)

X(14360) lies on the cubics K008, K301, the anticomplement of the circumcircle, and these lines: {2, 99}, {4, 10748}, {5, 11258}, {20, 1296}, {23, 5866}, {30, 5971}, {67, 69}, {75, 149}, {145, 10704}, {146, 2780}, {147, 2793}, {150, 2813}, {151, 2819}, {152, 2824}, {153, 2830}, {193, 10765}, {305, 1369}, {316, 3266}, {325, 1272}, {339, 7386}, {376, 6031}, {388, 3325}, {497, 6019}, {542, 9146}, {858, 6390}, {1370, 13219}, {1494, 10513}, {2847, 6527}, {3048, 9544}, {3091, 5512}, {3146, 10734}, {3616, 11721}, {5169, 11671}, {5468, 9143}, {5485, 10354}, {5913, 9870}, {5969, 6792}, {6077, 6082}, {7417, 13172}, {7533, 11059}, {8030, 10488}, {8593, 10552}, {8716, 9745}, {9129, 10330}, {9541, 11835}, {9542, 11833}

X(14360) = reflection of X(i) in X(j) for these {i,j}: {2, 10717}, {4, 10748}, {20, 1296}, {111, 126}, {145, 10704}, {149, 10779}, {193, 10765}, {3146, 10734}, {6082, 6077}, {9870, 5913}, {11258, 5}
X(14360) = isotomic conjugate of X(13574)
X(14360) = anticomplement X(111)
X(14360) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {1, 524}, {57, 4442}, {63, 858}, {75, 316}, {162, 9979}, {187, 192}, {468, 5905}, {524, 8}, {662, 690}, {896, 2}, {897, 671}, {922, 194}, {1581, 11646}, {1910, 10754}, {2349, 9140}, {2642, 148}, {3266, 6327}, {3292, 6360}, {3712, 329}, {4062, 2895}, {4235, 7253}, {4750, 149}, {5467, 4560}, {5468, 7192}, {6390, 4329}, {6629, 75}, {7181, 7}, {9395, 1648}, {14210, 69}
X(14360) = X(i)-Ceva conjugate of X(j) for these (i,j): {316, 69}, {3266, 2}, {5189, 1369}
X(14360) = orthoptic-circle-of-Steiner-inellipse inverse of X(620)
X(14360) = orthoptic-circle-of-Steiner-circumellipse inverse of X(99)
X(14360) = DeLongchamps-circle inverse of X(2373)
X(14360) = isoconjugate of X(31) and X(13574)
X(14360) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (111, 126, 2), (111, 10717, 126)
X(14360) = barycentric product X(76)*X(2930)
X(14360) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 13574}, {2930, 6}

X(14361) =  ISOTOMIC CONJUGATE OF X(1032)

Barycentrics    (tan A)(tan^2 A - tan^2 B - tan^2 C) : :
Barycentrics    (a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*(a^8 - 4*a^6*b^2 + 6*a^4*b^4 - 4*a^2*b^6 + b^8 - 4*a^6*c^2 - 4*a^4*b^2*c^2 + 4*a^2*b^4*c^2 + 4*b^6*c^2 + 6*a^4*c^4 + 4*a^2*b^2*c^4 - 10*b^4*c^4 - 4*a^2*c^6 + 4*b^2*c^6 + c^8) : :

X(14361) lies on the cubic K007 and these lines: {2, 253}, {4, 51}, {7, 92}, {8, 1034}, {20, 3183}, {69, 1032}, {107, 3079}, {297, 6392}, {324, 6819}, {329, 8894}, {393, 13567}, {1033, 6527}, {1056, 1148}, {1058, 7049}, {1370, 12384}, {1498, 6523}, {1503, 6525}, {1853, 10002}, {2883, 6526}, {3462, 5067}, {3535, 13886}, {3536, 13939}, {4176, 6331}, {5056, 8888}, {5656, 6624}, {5667, 11001}, {6515, 6820}, {6530, 6619}, {6618, 6776}, {8743, 10601}

X(14361) = isotomic conjugate of X(1032)
X(14361) = complement of X(20213)
X(14361) = anticomplement X(1073)
X(14361) = polar conjugate of X(3346)
X(14361) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {1, 253}, {19, 3146}, {20, 4329}, {154, 6360}, {162, 3265}, {204, 2}, {610, 20}, {1249, 8}, {1394, 347}, {1895, 69}, {3172, 192}, {3198, 3151}, {3213, 145}, {4183, 2184}, {5930, 2897}, {6525, 5905}, {7156, 144}
X(14361) = X(i)-Ceva conjugate of X(j) for these (i,j): {69, 4}, {6527, 6616}
X(14361) = X(i)-cross conjugate of X(j) for these (i,j): {1033, 6523}, {1498, 6527}, {3343, 2}
X(14361) = cevapoint of X(i) and X(j) for these (i,j): {1033, 1498}, {1249, 3183}, {3176, 8894}
X(14361) = perspector of ABC and pedal triangle of X(3183)
X(14361) = perspector of ABC and the reflection in X(1249) of the pedal triangle of X(1249)
X(14361) = perspector of pedal and antipedal triangles of X(20)
X(14361) = concurrence of extraversions of line X(7)X(92)
X(14361) = cyclocevian conjugate of X(14365)
X(14361) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (107, 11206, 3079), (196, 7003, 92), (459, 1249, 2), (1498, 6523, 6616), (1899, 6524, 4), (2052, 11433, 4), (3176, 7149, 1895), (12324, 14249, 4)
X(14361) = X(i)-isoconjugate of X(j) for these (i,j): {31, 1032}, {48, 3346}, {212, 8810}, {603, 8805}
X(14361) = barycentric product X(i)*X(j) for these {i,j}: {4, 6527}, {69, 6523}, {75, 1712}, {76, 1033}, {253, 6616}, {264, 1498}, {2052, 6617}
X(14361) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 1032}, {4, 3346}, {278, 8810}, {281, 8805}, {1033, 6}, {1249, 3344}, {1498, 3}, {1712, 1}, {3183, 3350}, {3343, 1073}, {3349, 3348}, {6523, 4}, {6527, 69}, {6616, 20}, {6617, 394}, {8803, 73}, {8807, 1214}, {8886, 1433}

X(14362) =  ANTICOMPLEMENT OF X(3344)

Trilinears    b^2c^2(2x^3yz - x^2y^2 - x^2z^2 + y^2z^2 - x^2y^2z^2)/(x - yz) : :, where x : y : z = cos A : cos B : cos C
Barycentrics   (a^4 + 2*a^2*b^2 - 3*b^4 - 2*a^2*c^2 + 2*b^2*c^2 + c^4)*(-a^4 + 2*a^2*b^2 - b^4 - 2*a^2*c^2 - 2*b^2*c^2 + 3*c^4)*(5*a^12 - 10*a^10*b^2 - 9*a^8*b^4 + 36*a^6*b^6 - 29*a^4*b^8 + 6*a^2*b^10 + b^12 - 10*a^10*c^2 + 34*a^8*b^2*c^2 - 36*a^6*b^4*c^2 + 4*a^4*b^6*c^2 + 14*a^2*b^8*c^2 - 6*b^10*c^2 - 9*a^8*c^4 - 36*a^6*b^2*c^4 + 50*a^4*b^4*c^4 - 20*a^2*b^6*c^4 + 15*b^8*c^4 + 36*a^6*c^6 + 4*a^4*b^2*c^6 - 20*a^2*b^4*c^6 - 20*b^6*c^6 - 29*a^4*c^8 + 14*a^2*b^2*c^8 + 15*b^4*c^8 + 6*a^2*c^10 - 6*b^2*c^10 + c^12) : :

X(14362) lies on the cubic K007 and thesse lines:
{2, 1032}, {4, 253}, {7, 1034}, {20, 2130}, {189, 2184}

X(14362) = isotomic conjugate of X(14365)
X(14362) = perspector of ABC and pedal triangle of X(2130)
X(14362) = cyclocevian conjugate of perspector of ABC and pedal triangle of X(2131)
X(14362) = midpoint of X(2130) and (Darboux image of X(2130))
X(14362) = isotomic conjugate of perspector of ABC and pedal triangle of X(3348)
X(14362) = anticomplement X(3344)
X(14362) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {1, 1032}, {2184, 12324}, {3343, 8}
X(14362) = X(69)-Ceva conjugate of X(253)
X(14362) = X(3350)-cross conjugate of X(2)
X(14362) = cevapoint of X(2130) and X(3343)
X(14362) = isoconjugate of X(204) and X(3348)
X(14362) = barycentric quotient X(i)/X(j) for these {i,j}: {1073, 3348}, {3183, 1249}, {3343, 3349}, {3350, 3344}, {3356, 2131}

X(14363) =  MIDPOINT OF X(1075) AND X(14249)

Barycentrics   (a^2+b^2-c^2) (a^2-b^2+c^2) (a^2 b^2-b^4+a^2 c^2+2 b^2 c^2-c^4) (a^8-3 a^6 b^2+3 a^4 b^4-a^2 b^6-3 a^6 c^2+a^2 b^4 c^2+2 b^6 c^2+3 a^4 c^4+a^2 b^2 c^4-4 b^4 c^4-a^2 c^6+2 b^2 c^6) : :
X(14363) = X(4) + 3 X(1075) = 5 X(1656) - 3 X(14059) = X(4) - 3 X(14249)

X(14363) lies on the cubics K026, K671, and these lines: {4, 51}, {5, 8798}, {107, 10282}, {133, 2883}, {550, 3184}, {800, 1990}, {1656, 14059}, {6525, 9833}

X(14363) = midpoint of X(1075) and X(14249)
X(14363) = reflection of X(8798) in X(5)
X(14363) = X(i)-complementary conjugate of X(j) for these (i,j): {31, 53}, {6759, 10}
X(14363) = X(2)-Ceva conjugate of X(53)
X(14363) = X(8798)-of-Johnson-triangle
X(14363) = barycentric product X(324)X(6759)
X(14363) = barycentric quotient X(6759)/X(97)
X(14363) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (51, 13450, 8887), (1093, 3168, 389)

X(14364) =  ISOTOMIC CONJUGATE OF X(11061)

Barycentrics   (a^8-2 a^4 b^4+b^8-2 a^6 c^2+a^4 b^2 c^2+a^2 b^4 c^2-2 b^6 c^2+2 a^4 c^4-3 a^2 b^2 c^4+2 b^4 c^4+2 a^2 c^6+2 b^2 c^6-3 c^8) (a^8-2 a^6 b^2+2 a^4 b^4+2 a^2 b^6-3 b^8+a^4 b^2 c^2-3 a^2 b^4 c^2+2 b^6 c^2-2 a^4 c^4+a^2 b^2 c^4+2 b^4 c^4-2 b^2 c^6+c^8) : :

X(14364) is the perspector on the Droussent cubic (K008) corresponding to X(10417) on the Droussent central cubic (K042). (Randy Hutson, November 2, 2017)

X(14364) lies on the cubic K008 and these lines: {316, 10417}, {524, 5523}, {858, 6390}

X(14364) = isotomic conjugate of X(11061)
X(14364) = X(67)-cross conjugate of X(2)
X(14364) = antigonal image of X(10415)
X(14364) = X(i)-isoconjugate of X(j) for these (i,j): {31, 11061}, {922, 10416}
X(14364) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 11061}, {671, 10416}, {10417, 187}

X(14365) =  ANTICOMPLEMENT OF X(3350)

Trilinears    (x - yz)/(2x^3yz - x^2y^2 - x^2z^2 + y^2z^2 - x^2y^2z^2) : : , where x : y : z = cos A : cos B : cos C
Barycentrics    (3*a^4 - 2*a^2*b^2 - b^4 - 2*a^2*c^2 + 2*b^2*c^2 - c^4)*(a^12 + 6*a^10*b^2 - 29*a^8*b^4 + 36*a^6*b^6 - 9*a^4*b^8 - 10*a^2*b^10 + 5*b^12 - 6*a^10*c^2 + 14*a^8*b^2*c^2 + 4*a^6*b^4*c^2 - 36*a^4*b^6*c^2 + 34*a^2*b^8*c^2 - 10*b^10*c^2 + 15*a^8*c^4 - 20*a^6*b^2*c^4 + 50*a^4*b^4*c^4 - 36*a^2*b^6*c^4 - 9*b^8*c^4 - 20*a^6*c^6 - 20*a^4*b^2*c^6 + 4*a^2*b^4*c^6 + 36*b^6*c^6 + 15*a^4*c^8 + 14*a^2*b^2*c^8 - 29*b^4*c^8 - 6*a^2*c^10 + 6*b^2*c^10 + c^12)*(a^12 - 6*a^10*b^2 + 15*a^8*b^4 - 20*a^6*b^6 + 15*a^4*b^8 - 6*a^2*b^10 + b^12 + 6*a^10*c^2 + 14*a^8*b^2*c^2 - 20*a^6*b^4*c^2 - 20*a^4*b^6*c^2 + 14*a^2*b^8*c^2 + 6*b^10*c^2 - 29*a^8*c^4 + 4*a^6*b^2*c^4 + 50*a^4*b^4*c^4 + 4*a^2*b^6*c^4 - 29*b^8*c^4 + 36*a^6*c^6 - 36*a^4*b^2*c^6 - 36*a^2*b^4*c^6 + 36*b^6*c^6 - 9*a^4*c^8 + 34*a^2*b^2*c^8 - 9*b^4*c^8 - 10*a^2*c^10 - 10*b^2*c^10 + 5*c^12) : :

X(14365) lies on the cubic K007 and these lines: {2, 3349}, {20, 3183}, {69, 14362}, {329, 34162}, {2060, 28781}, {5932, 8812}, {6616, 36413}

X(14365) = isogonal conjugate of X(28785)
X(14365) = isotomic conjugate of X(14362)
X(14365) = polar conjugate of X(40839)
X(14365) = cyclocevian conjugate of X(14361)
X(14365) = anticomplement of the isogonal conjugate of X(3349)
X(14365) = isotomic conjugate of the anticomplement of X(3344)
X(14365) = isotomic conjugate of the isogonal conjugate of X(28781)
X(14365) = X(3349)-anticomplementary conjugate of X(8)
X(14365) = X(i)-cross conjugate of X(j) for these (i,j): {4, 20}, {3344, 2}
X(14365) = cevapoint of X(3348) and X(3349)
X(14365) = perspector of ABC and pedal triangle of X(3348)
X(14365) = X(i)-isoconjugate of X(j) for these (i,j): {1, 28785}, {31, 14362}, {48, 40839}, {3183, 19614}
X(14365) = barycentric product X(i)*X(j) for these {i,j}: {76, 28781}, {3348, 15466}, {14615, 31956}
X(14365) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 14362}, {4, 40839}, {6, 28785}, {1249, 3183}, {2131, 3356}, {3344, 3350}, {3348, 1073}, {3349, 3343}, {14481, 2130}, {28781, 6}, {31956, 64}, {36413, 2060}, {36908, 8812}
X(14365) = {X(3349),X(31943)}-harmonic conjugate of X(2)

X(14366) =  CIRCUMCIRCLE-INVERSE OF X(249)

Barycentrics   a^2*(a^2 - b^2)^2*(a^2 - c^2)^2*(-a^6 + a^4*b^2 - a^2*b^4 + b^6 + a^4*c^2 + a^2*b^2*c^2 - b^4*c^2 - a^2*c^4 - b^2*c^4 + c^6) : :

X(14366) lies on the cubics K072, I568, K845 and these lines: {3, 249}, {4, 250}, {501, 1101}, {4590, 7768}

X(14366) = isogonal conjugate of X(6328)
X(14366) = anticomplement of X(33967)
X(14366) = antigonal image of X(3448)
X(14366) = circumcircle-inverse of X(249)
X(14366) = X(99)-Ceva conjugate of X(249)
X(14366) = syngonal conjugate of X(110)
X(14366) = Cundy-Parry Phi transform of X(14355)
X(14366) = Cundy-Parry Psi transform of X(14356)
X(14366) = X(i)-isoconjugate of X(j) for these (i,j): {1, 6328}, {1109, 3447}, {2643, 13485}
X(14366) = barycentric product X(i)*X(j) for these {i,j}: {249, 3448}, {4590, 7669}
X(14366) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 6328}, {249, 13485}, {3448, 338}, {7669, 115}, {8574, 8029}

X(14367) =  CIRCUMCIRCLE-INVERSE OF X(1263)

Barycentrics   a^2*(-a^6 + a^4*b^2 + a^2*b^4 - b^6 + 3*a^4*c^2 - a^2*b^2*c^2 + 3*b^4*c^2 - 3*a^2*c^4 - 3*b^2*c^4 + c^6)*(a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6 - a^4*c^2 + a^2*b^2*c^2 + 3*b^4*c^2 - a^2*c^4 - 3*b^2*c^4 + c^6)*(a^8 - 4*a^6*b^2 + 6*a^4*b^4 - 4*a^2*b^6 + b^8 - 4*a^6*c^2 + 5*a^4*b^2*c^2 + a^2*b^4*c^2 - 2*b^6*c^2 + 6*a^4*c^4 + a^2*b^2*c^4 + 2*b^4*c^4 - 4*a^2*c^6 - 2*b^2*c^6 + c^8) : :

X(14367) lies on the cubics K073, K439, K467 and these lines: {3, 1263}, {5, 10227}, {54, 1511}, {115, 2963}, {140, 10277}, {1291, 2070}, {2937, 12011}

X(14367) = isogonal conjugate of X(11584)
X(14367) = crosspoint of X(195) and X(8494)
X(14367) = crosssum of X(3459) and X(5684)
X(14367) = barycentric product X(195)X(13582)
X(14367) = barycentric quotient X(6)/X(11584)
X(14367) = X(i)-isoconjugate of X(j) for these (i,j): {1, 11584}, {1749, 3459}

X(14368) =  CIRCUMCIRCLE-INVERSE OF X(616)

Barycentrics   a^2 (a^2-b^2-b c-c^2) (a^2-b^2+b c-c^2) (Sqrt(3) (a^6-a^4 b^2+a^2 b^4-b^6-a^4 c^2+2 a^2 b^2 c^2+b^4 c^2+a^2 c^4+b^2 c^4-c^6)+2 (a^4-b^4+4 b^2 c^2-c^4) S) : :

X(14368) lies on the cubics K073 and K438a, and on these lines: {2, 6106}, {3, 299}, {15, 323}, {23, 5978}, {74, 10410}, {186, 340}, {533, 6105}, {617, 2926}, {619, 6104}, {621, 3129}

X(14368) = isogonal conjugate of X(14372)
X(14368) = anticomplement X(6106)
X(14368) = circumcircle-inverse of X(616)
X(14368) = X(299)-Ceva conjugate of X(323)
X(14368) = isoconjugate of X(2166) and X(3438)
X(14368) = barycentric product X(i)*X(j) for these {i,j}: {323, 621}, {3129, 7799}
X(14368) = barycentric quotient X(i)/X(j) for these {i,j}: {50, 3438}, {323, 2992}, {621, 94}, {3129, 1989}
X(14368) = {X(616),X(628)}-harmonic conjugate of X(2992)

X(14369) =  CIRCUMCIRCLE-INVERSE OF X(617)

Barycentrics    a^2 (a^2-b^2-b c-c^2) (a^2-b^2+b c-c^2) (Sqrt(3) (a^6-a^4 b^2+a^2 b^4-b^6-a^4 c^2+2 a^2 b^2 c^2+b^4 c^2+a^2 c^4+b^2 c^4-c^6)-2 (a^4-b^4+4 b^2 c^2-c^4) S) : :

X(14369) lies on the cubics K073 and K438b, and on these lines: {2,6107}, {3,298}, {16,323}, {23,5979}, {74,10409}, {186,340}, {532,6104}, {616,2925}, {618,6105}, {622,3130}

X(14369) = isogonal conjugate of X(14373)
X(14369) = anticomplement X(6107)
X(14369) = circumcircle-inverse of X(617)
X(14369) = X(298)-Ceva conjugate of X(323)
X(14369) = isoconjugate of X(2166) and X(3439)
X(14369) = barycentric product X(i)*X(j) for these {i,j}: {323, 622}, {3130, 7799}
X(14369) = barycentric quotient X(i)/X(j) for these {i,j}: {50, 3439}, {323, 2993}, {622, 94}, {3130, 1989}
X(14369) = {X(616),X(628)}-harmonic conjugate of X(2992)
X(14369) = {X(617),X(627)}-harmonic conjugate of X(2993)

X(14370) =  ISOGONAL CONJUGATE OF X(2896)

Barycentrics    a^2 (a^4+a^2 b^2+b^4+a^2 c^2+b^2 c^2-c^4) (a^4+a^2 b^2-b^4+a^2 c^2+b^2 c^2+c^4) : :

The trilinear polar of X(14370) passes through X(3005). (Randy Hutson, November 2, 2017)

Let Ra be the radical axis of the 1st Brocard circle and A-reflected-Neuberg circle, and define Rb and Rc cyclically. Let A' = Rb∩Rc, and define B'and C' cyclically. The lines AA', BB', CC' concur in X(14370). (Randy Hutson, November 2, 2017)

X(14370) lies on the cubics K020 and K655, and on these lines: {39, 1915}, {141, 384}, {1625, 3493}, {1843, 11380}, {3224, 8874}, {3494, 8866}, {3496, 3954}, {3498, 8861}, {3503, 8867}, {7797, 9484}, {8790, 8864}, {9481, 9482}

X(14370) = isogonal conjugate of X(2896)
X(14370) = X(i)-cross conjugate of X(j) for these (i,j): {251, 6}, {695, 3224}
X(14370) = crosssum of X(2) and X(8272)
X(14370) = Steiner image of X(6)
X(14370) = trilinear pole of radical axis of circumcircle and symmedial circle
X(14370) = barycentric product X(6)X(1031)
X(14370) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 2896}, {32, 10329}, {1031, 76}
X(14370) = X(i)-isoconjugate of X(j) for these (i,j): {1, 2896}, {75, 10329}

X(14371) =  X(6)-CROSS CONJUGATE OF X(97)

Barycentrics    a^2 (a^2-b^2-c^2) (a^4-2 a^2 b^2+b^4-a^2 c^2-b^2 c^2) (a^4-a^2 b^2-2 a^2 c^2-b^2 c^2+c^4) (a^6 b^2-3 a^4 b^4+3 a^2 b^6-b^8-2 a^6 c^2-a^4 b^2 c^2+3 b^6 c^2+4 a^4 c^4-a^2 b^2 c^4-3 b^4 c^4-2 a^2 c^6+b^2 c^6) (2 a^6 b^2-4 a^4 b^4+2 a^2 b^6-a^6 c^2+a^4 b^2 c^2+a^2 b^4 c^2-b^6 c^2+3 a^4 c^4+3 b^4 c^4-3 a^2 c^6-3 b^2 c^6+c^8) : :

X(14371) lies on the cubic K361 and these lines: {97, 6759}, {275, 13855}, {1294, 8884}

X(14371) = X(6)-cross conjugate of X(97)

X(14372) =  ISOGONAL CONJUGATE OF X(14368)

Barycentrics    (-a^2+b^2-a c-c^2) (-a^2+b^2+a c-c^2) (-a^2-a b-b^2+c^2) (-a^2+a b-b^2+c^2) (Sqrt(3) (-a^6+a^4 b^2-a^2 b^4+b^6+a^4 c^2+2 a^2 b^2 c^2-b^4 c^2+a^2 c^4+b^2 c^4-c^6)+2 (-a^4+b^4+4 a^2 c^2-c^4) S) (Sqrt(3) (-a^6+a^4 b^2+a^2 b^4-b^6+a^4 c^2+2 a^2 b^2 c^2+b^4 c^2-a^2 c^4-b^2 c^4+c^6)+2 (-a^4+4 a^2 b^2-b^4+c^4) S) : :

X(14372) lies on the cubic K060 and these lines: {5,11119}, {13,1605}, {30,1338}, {265,300}, {8737,11060}, {11080,11088}, {11082,11086}

X(14372) = isogonal conjugate of X(14368)
X(14372) = antigonal image of X(3440)
X(14372) = X(3458)-cross conjugate of X(1989)
X(14372) = X(i)-isoconjugate of X(j) for these (i,j): {1, 14638}, {621, 6149}
X(14372) = barycentric product X(i)*X(j) for these {i,j}: {94, 3438}, {1989, 2992}
X(14372) = barycentric quotient X(i)/X(j) for these {i,j}: {6,14368}, {1989, 621}, {2992, 7799}, {3438, 323}, {8737, 11093}, {11060, 3129}, {11089, 1338}

X(14373) =  ISOGONAL CONJUGATE OF X(14369)

Barycentrics    (-a^2+b^2-a c-c^2) (-a^2+b^2+a c-c^2) (-a^2-a b-b^2+c^2) (-a^2+a b-b^2+c^2) (Sqrt(3) (-a^6+a^4 b^2-a^2 b^4+b^6+a^4 c^2+2 a^2 b^2 c^2-b^4 c^2+a^2 c^4+b^2 c^4-c^6)-2 (-a^4+b^4+4 a^2 c^2-c^4) S) (Sqrt(3) (-a^6+a^4 b^2+a^2 b^4-b^6+a^4 c^2+2 a^2 b^2 c^2+b^4 c^2-a^2 c^4-b^2 c^4+c^6)-2 (-a^4+4 a^2 b^2-b^4+c^4) S) : :

X(14373) lies on the cubic K060 and these lines: {5,11120}, {14,1606}, {30,1337}, {265,301}, {8738,11060}, {11081,11087}, {11083,11085}

X(14373) = isogonal conjugate of X(14369)
X(14373) = antigonal image of X(3441)
X(14373) = X(i)-isoconjugate of X(j) for these (i,j): {1, 14369}, {622, 6149}
X(14373) = barycentric product X(i)*X(j) for these {i,j}: {94, 3439}, {1989, 2993}
X(14373) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 14369}, {1989, 622}, {2993, 7799}, {3439, 323}, {8738, 11094}, {11060, 3130}, {11084, 1337}

X(14374) =  X(5)X(2574)∩X(30)X(143)

Barycentrics    a^2*(a^4 - a^2*b^2 - 2*a^2*c^2 - b^2*c^2 + c^4 - a^2*b^2*J + b^4*J - b^2*c^2*J)*(a^4 - 2*a^2*b^2 + b^4 - a^2*c^2 - b^2*c^2 - a^2*c^2*J - b^2*c^2*J + c^4*J) : :

X(14374) lies on the Jerabek hyperbola, the cubics K054 and K714, and these lines: {5, 2574}, {30, 143}, {54, 1113}, {64, 1345}, {185, 2575}, {3521, 10751}

X(14374) = reflection of X(14375) in X(13630)
X(14374) = X(1312)-cross conjugate of X(2575)

X(14375) =  X(5)X(2575)∩X(30)X(143)

Barycentrics    a^2*(a^4 - a^2*b^2 - 2*a^2*c^2 - b^2*c^2 + c^4 + a^2*b^2*J - b^4*J + b^2*c^2*J)*(a^4 - 2*a^2*b^2 + b^4 - a^2*c^2 - b^2*c^2 + a^2*c^2*J + b^2*c^2*J - c^4*J) : :

X(14375) lies on the Jerabek hyhperbola, the cubics K054 and K714, and the lines {5, 2575}, {30, 143}, {54, 1114}, {64, 1344}, {185, 2574}, {3521, 10750}

X(14375) = reflection of X(14374) in X(13630)
X(14375) = X(1313)-cross conjugate of X(2574)

X(14376) =  X(3)X(66)∩X(4)X(127)

Barycentrics    (a^2-b^2-c^2) (a^4+b^4-c^4) (a^4-b^4+c^4) : :

X(14376) lies on the cubics K009, K527, K836 and these lines: {2, 1235}, {3, 66}, {4, 127}, {69, 10316}, {216, 7822}, {339, 3767}, {394, 441}, {577, 7794}, {631, 5481}, {1073, 9605}, {1217, 7404}, {2972, 14003}, {3088, 3346}, {3095, 14059}, {3284, 7855}, {3548, 3788}, {3549, 3934}, {5158, 7889}, {6760, 12054}, {7494, 10130}, {7778, 11585}, {7784, 12605}, {10317, 14023}

X(14376) = isogonal conjugate of X(8743)
X(14376) = isotomic conjugate of polar conjugate of X(66)
X(14376) = X(48)-complementary conjugate of X(3162)
X(14376) = X(1289)-Ceva conjugate of X(525)
X(14376) = X(i)-cross conjugate of X(j) for these (i,j): {39, 3}, {184, 69}
X(14376) = X(i)-isoconjugate of X(j) for these (i,j): {1, 8743}, {4, 2172}, {19, 22}, {25, 1760}, {28, 4456}, {92, 206}, {158, 10316}, {162, 2485}, {240, 11610}, {273, 4548}, {315, 1973}, {318, 7251}, {607, 7210}, {608, 4123}, {1474, 4463}, {2203, 4150}
X(14376) = trilinear pole of line {520, 2525}
X(14376) = {X(6389),X(7795)}-harmonic conjugate of X(3)
X(14376) = Cundy-Parry Phi transform of X(1503)
X(14376) = Cundy-Parry Psi transform of X(1297)
X(14376) = crosssum of X(6) and X(3162)
X(14376) = barycentric product X(i)*X(j) for these {i,j}: {66, 69}, {304, 2156}, {305, 2353}, {1289, 3265}, {3926, 13854}
X(14376) = barycentric quotient X(i)/X(j) for these {i,j}: {3, 22}, {6, 8743}, {48, 2172}, {63, 1760}, {66, 4}, {67, 11605}, {69, 315}, {71, 4456}, {72, 4463}, {77, 7210}, {78, 4123}, {184, 206}, {248, 11610}, {306, 4150}, {520, 8673}, {577, 10316}, {647, 2485}, {1289, 107}, {2156, 19}, {2353, 25}, {3917, 3313}, {4558, 4611}, {13854, 393}

X(14377) =  X(3)X(142)∩X(4)X(103)

Barycentrics    (a^2+a b+b^2-a c-b c) (a^2-a b+a c-b c+c^2) : :

X(14377) lies on the cubic K009 and these lines: {2, 1796}, {3, 142}, {4, 103}, {7, 1803}, {32, 1086}, {58, 4000}, {63, 169}, {85, 5011}, {101, 4209}, {222, 553}, {277, 3474}, {295, 2809}, {481, 6502}, {482, 2067}, {544, 6604}, {596, 4361}, {673, 4253}, {1790, 8025}, {1797, 5773}, {2141, 8049}, {3423, 4292}, {3579, 6706}, {4904, 7354}, {5903, 9317}

X(14377) = isogonal conjugate of X(3730)
X(14377) = X(i)-cross conjugate of X(j) for these (i,j): {1475, 1}, {1565, 514}, {2308, 86}, {11246, 7}
X(14377) = X(i)-isoconjugate of X(j) for these (i,j): {1, 3730}, {6, 3681}, {37, 4184}, {58, 4006}, {100, 6586}, {101, 1734}, {116, 1110}
X(14377) = cevapoint of X(i) and X(j) for these (i,j): {649, 1086}, {4466, 4988}
X(14377) = trilinear pole of line {676, 1459}
X(14377) = Cundy-Parry Phi transform of X(516)
X(14377) = Cundy-Parry Psi transform of X(103)
X(14377) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 3681}, {6, 3730}, {37, 4006}, {58, 4184}, {513, 1734}, {649, 6586}, {1086, 116}

X(14378) =  ISOGONAL CONJUGATE OF X(14247)

Barycentrics    (b^2+c^2) (a^4+a^2 b^2+b^4-c^4) (-a^4+b^4-a^2 c^2-c^4) : :

X(14378) lies on the cubic K009 and these lines: {3, 2916}, {305, 7871}

X(14378) = isogonal conjugate of X(14247)
X(14378) = X(11205)-cross conjugate of X(141)
X(14378) = X(i)-isoconjugate of X(j) for these (i,j): {1, 14247}, {82, 6636}
X(14378) = barycentric product X(3456)X(8024)
X(14378) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 14247}, {39, 6636}, {141, 7768}, {3456, 251}

X(14379) =  X(3)X(64)∩X(4)X(122)

Trilinears    (cos^2 A)/(cos A - cos B cos C) : :
Barycentrics    a^4 SA^2/(SA*SB + SA*SC - SB*SC) : :      (Paul Yiu, Hyacinthos #21973 4/17/2013)
Barycentrics    a^4*(a^2 - b^2 - c^2)^2*(a^4 - 2*a^2*b^2 + b^4 + 2*a^2*c^2 + 2*b^2*c^2 - 3*c^4)*(a^4 + 2*a^2*b^2 - 3*b^4 - 2*a^2*c^2 + 2*b^2*c^2 + c^4) : :

X(14379) is the perspector of the circumconic concentric and homothetic to the following conic described by Paul Yiu in response to a problem posed by Randy Hutson (Hyacinthos #21973 and related, 4/17/2013):

Let A'B'C' be the pedal triangle of X(1498) (which is also the cevian triangle of X(20)). Let Ba, Ca be the orthogonal projections of A' onto lines CA, AB, resp. Define Cb, Ab, Ac, Bc cyclically. The points Ba, Ca, Cb, Ab, Ac, Bc lie on a conic, here named the Yiu-Hutson conic. Note: If X(4) is substituted for X(1498), the conic is the Taylor circle. The Yiu-Hutson conic has center X(14390) and is given (Paul Yiu, Hyacinthos #21973 4/17/2013) by this barycentric equation:

(4/S^2) cyclic sum ((a^4 SA^2)/(SA*SB + SA*SC -SB*SC}))yz - (x + y + z)^2 = 0

An alternative construction for points Ba, Ca, Cb, Ab, Ac, Bc above follows: Let A'B'C' be the half-altitude triangle. Then Ab = BC∩C'A', Ac = BC∩A'B', and Bc, Ba, Ca, Cb are defined cyclically. Note: The centroid of BaCaCbAbAcBc is X(2). (Randy Hutson, September 10, 2017)

X(14379) lies on the cubics K009, K576, and these lines: {2, 1105}, {3, 64}, {4, 122}, {95, 253}, {212, 7114}, {378, 5879}, {417, 577}, {459, 631}, {549, 13157}, {578, 13855}, {1204, 2972}, {3343, 11425}, {3346, 6525}, {6337, 6394}

X(14379) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 64, 11589), (3, 1073, 64), (3, 6760, 6759), (3, 14059, 3357), (64, 1073, 8798), (8798, 11589, 64)
X(14379) = isogonal conjugate of X(14249)
X(14379) = X(i)-Ceva conjugate of X(j) for these (i,j): (2,14390), (1301,520)
X(14379) = X(184)-cross conjugate of X(577)
X(14379) = crosssum of X(i) and X(j) for these (i,j): {4, 6523}, {1249, 6525}
X(14379) = X(i)-isoconjugate of X(j) for these (i,j): {1, 14249}, {4, 1895}, {20, 158}, {75, 6525}, {92, 1249}, {204, 264}, {331, 7156}, {610, 2052}, {821, 2883}, {823, 6587}, {1097, 6526}, {1784, 10152}, {1896, 5930}, {1969, 3172}, {3213, 7017}
X(14379) = X(6759)-vertex conjugate of X(13855)
X(14379) = Cundy-Parry Phi transform of X(6000)
X(14379) = Cundy-Parry Psi transform of X(1294)
X(14379) = barycentric product X(i)*X(j) for these {i,j}: {3, 1073}, {64, 394}, {97, 8798}, {253, 577}, {255, 2184}, {326, 2155}, {459, 1092}, {2289, 8809}
X(14379) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 14249}, {32, 6525}, {48, 1895}, {64, 2052}, {184, 1249}, {577, 20}, {1073, 264}, {2155, 158}, {4055, 8804}, {8798, 324}, {9247, 204}

X(14380) =  X(3)X(520)∩X(4)X(523)

Barycentrics    a^2 (b^2-c^2) (a^2-b^2-c^2) (a^4-2 a^2 b^2+b^4+a^2 c^2+b^2 c^2-2 c^4) (a^4+a^2 b^2-2 b^4-2 a^2 c^2+b^2 c^2+c^4) : :

X( 14380) lies on the Jerabek hyperbola, the cubics K027, K074, K229, and these lines: {3, 520}, {4, 523}, {6, 647}, {54, 8562}, {64, 924}, {67, 9003}, {68, 8057}, {69, 3265}, {74, 526}, {125, 14220}, {265, 6334}, {290, 1494}, {512, 3426}, {525, 4846}, {684, 895}, {690, 10293}, {1173, 3470}, {1304, 5502}, {2492, 8749}, {3134, 12079}, {3521, 6368}, {5504, 8552}, {5627, 10412}, {6086, 10152}, {9139, 9178}

X(14380) = reflection of X(14220) in X(125)
X(14380) = isogonal conjugate of X(4240)
X(14380) = antigonal image of X(14220)
X(14380) = X(i)-zayin conjugate of X(j) for these (i,j): {1, 4240}, {1021, 2173}
X(14380) = X(i)-Ceva conjugate of X(j) for these (i,j): {1304, 74}, {2394, 2433}, {5627, 125}
X(14380) = X(i)-cross conjugate of X(j) for these (i,j): {686, 525}, {9409, 647}
X(14380) = X(i)-isoconjugate of X(j) for these (i,j): {1, 4240}, {19, 2407}, {30, 162}, {92, 2420}, {110, 1784}, {112, 14206}, {648, 2173}, {662, 1990}, {811, 1495}, {823, 3284}, {1099, 1304}, {5379, 11125}, {6331, 9406}
X(14380) = cevapoint of X(647) and X(9409)
X(14380) = crosspoint of X(74) and X(1304)
X(14380) = trilinear pole of line {647, 3269}
X(14380) = crossdifference of every pair of points on line {30, 1990}
X(14380) = crosssum of X(i) and X(j) for these (i,j): {30, 9033}, {523, 7687}, {1495, 1637}
X(14380) = barycentric product X(i)*X(j) for these {i,j}: {3, 2394}, {69, 2433}, {74, 525}, {647, 1494}, {656, 2349}, {2159, 14208}, {3265, 8749}, {3268, 11079}, {4558, 12079}, {5627, 8552}, {6334, 10419}
X(14380) = barycentric quotient X(i)/X(j) for these {i,j}: {3, 2407}, {6, 4240}, {74, 648}, {184, 2420}, {512, 1990}, {520, 11064}, {525, 3260}, {647, 30}, {656, 14206}, {661, 1784}, {686, 113}, {810, 2173}, {1494, 6331}, {2159, 162}, {2349, 811}, {2394, 264}, {2433, 4}, {2631, 1099}, {3049, 1495}, {3269, 9033}, {3284, 3233}, {6137, 6110}, {6138, 6111}, {8552, 6148}, {8749, 107}, {9409, 3163}, {9717, 4235}, {10097, 9214}, {10419, 687}, {11079, 476}

X(14381) =  ISOGONAL CONJUGATE OF X(14250)

Barycentrics    (2*a^2 + b^2 + c^2)*(a^4 + 3*a^2*b^2 + b^4 - c^4)*(a^4 - b^4 + 3*a^2*c^2 + c^4) : :

X(14381 lies on the cubic K009 and this line: {3, 7889}

X(14381) = isogonal conjugate of X(14250)

X(14382) =  X(3)X(76)∩X(4)X(2679)

Barycentrics    b^2 c^2 (-a^2+b c) (a^2+b c) (a^4+b^4-a^2 c^2-b^2 c^2) (-a^4+a^2 b^2+b^2 c^2-c^4) : :

X(14382) lies on the cubics K009, K056, K166, K512, K559 and these lines: {3, 76}, {4, 2679}, {32, 2966}, {83, 2422}, {248, 10342}, {880, 12215}, {3224, 6531}

X(14382) = isogonal conjugate of X(14251)
X(14382) = {X(76),X(14265)}-harmonic conjugate of X(290)
X(14382) = X(1956)-complementary conjugate of X(5031)
X(14382) = X(i)-cross conjugate of X(j) for these (i,j): {3978, 290}, {5027, 2966}
X(14382) = complement of the isogonal conjugate of X(32542)
X(14382) = X(i)-isoconjugate of X(j) for these (i,j): {1, 14251}, {237, 1581}, {325, 1927}, {511, 1967}, {694, 1755}, {1916, 9417}, {1934, 9418}, {1959, 9468}
X(14382) = X(98)-Hirst inverse of X(290)
X(14382) = trilinear pole of line {385, 14295}
X(14382) = Cundy-Parry Phi transform of X(2782)
X(14382) = Cundy-Parry Psi transform of X(2698)
X(14382) = barycentric product X(i)*X(j) for these {i,j}: {98, 3978}, {290, 385}, {880, 2395}, {1821, 1966}, {1910, 1926}, {2966, 14295}
X(14382) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 14251}, {98, 694}, {290, 1916}, {385, 511}, {419, 232}, {804, 3569}, {880, 2396}, {1580, 1755}, {1691, 237}, {1821, 1581}, {1910, 1967}, {1933, 9417}, {1966, 1959}, {1976, 9468}, {2395, 882}, {2422, 881}, {2966, 805}, {3978, 325}, {5026, 9155}, {5027, 2491}, {14295, 2799}

X(14383) = ISOGONAL CONJUGATE OF X(14252)

Barycentrics    (a^4 - a^2*b^2 - a^2*c^2 - 2*b^2*c^2)*(a^4*b^2 + a^2*b^4 + 2*a^4*c^2 + a^2*b^2*c^2 + 2*b^4*c^2 - 2*a^2*c^4 - 2*b^2*c^4)*(2*a^4*b^2 - 2*a^2*b^4 + a^4*c^2 + a^2*b^2*c^2 - 2*b^4*c^2 + a^2*c^4 + 2*b^2*c^4) : :

X(14383) lies on K009 and this line: {3, 3734}

X(14383) = isogonal conjugate of X(14252)

X(14384) =  ISOGONAL CONJUGATE OF X(14253)

Barycentrics    (2*a^4 - a^2*b^2 + b^4 - a^2*c^2 - 2*b^2*c^2 + c^4)*(a^6 + b^6 - 3*a^4*c^2 - a^2*b^2*c^2 - 3*b^4*c^2 + 3*a^2*c^4 + 3*b^2*c^4 - c^6)*(a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6 - a^2*b^2*c^2 + 3*b^4*c^2 - 3*b^2*c^4 + c^6) : :

X(14384) lies on the cubic K009 and this line: {3, 115}

X(14384) = isogonal conjugate of X(14253)

X(14385) =  X(3)X(74)∩X(4)X(447)

Barycentrics    a^4*(a^2 - b^2 - b*c - c^2)*(a^2 - b^2 + b*c - c^2)*(a^4 - 2*a^2*b^2 + b^4 + a^2*c^2 + b^2*c^2 - 2*c^4)*(a^4 + a^2*b^2 - 2*b^4 - 2*a^2*c^2 + b^2*c^2 + c^4) : :

X(14385) line on the cubic K009 and these lines: {2, 5627}, {3, 74}, {4, 477}, {24, 3447}, {54, 8562}, {140, 12079}, {186, 7740}, {566, 11079}, {1078, 1494}, {1650, 6760}, {5502, 14157}, {7577, 10421}

X(14385) = isogonal conjugate of X(14254)
X(14385) = X(6149)-complementary conjugate of X(11598)
X(14385) = X(i)-Ceva conjugate of X(j) for these (i,j): {1304, 526}, {10419, 74}
X(14385) = crosssum of X(30) and X(10272)
X(14385) = Cundy-Parry Phi transform of X(5663)
X(14385) = Cundy-Parry Psi transform of X(477)
X(14385) = X(i)-isoconjugate of X(i) and X(j) for these (i,j): {1, 14254}, {30, 2166}, {94, 2173}, {265, 1784}, {1099, 5627}, {1989, 14206}
X(14385) = trilinear product of vertices of anti-orthocentroidal triangle
X(14385) = barycentric product X(i)*X(j) for these {i,j}: {50, 1494}, {74, 323}, {1304, 8552}, {2349, 6149}, {2433, 10411}
X(14385) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 14254}, {50, 30}, {74, 94}, {323, 3260}, {2159, 2166}, {2433, 10412}, {6149, 14206}, {8749, 6344}, {14270, 1637}

X(14386) =  ISOGONAL CONJUGATE OF X(14255)

Barycentrics    a^4*(a^4 - 4*a^2*b^2 + b^4 + 5*a^2*c^2 + 5*b^2*c^2 - 8*c^4)*(a^4 + 5*a^2*b^2 - 8*b^4 - 4*a^2*c^2 + 5*b^2*c^2 + c^4)*(a^4 - 4*a^2*b^2 + b^4 - 4*a^2*c^2 + 5*b^2*c^2 + c^4) : :

X(14386) lies on the cubic K009 and these lines: {3, 2502}, {4, 9193}

X(14386) = isogonal conjugate of X(14255)
X(14386) = barycentric product X(9023)*X(9190)

X(14387) =  ISOTOMIC CONJUGATE OF X(3098)

Barycentrics    b^2 c^2 (-2 a^4+a^2 b^2+b^4-2 a^2 c^2+b^2 c^2-2 c^4) (2 a^4+2 a^2 b^2+2 b^4-a^2 c^2-b^2 c^2-c^4) : :

X(14387) lies on the cubic K056 and these lines: {30, 76}, {264, 1990}, {290, 7766}, {1502, 3260}, {9211, 9214}

X(14387) = isotomic conjugate of X(3098)
X(14387) = X(i)-isoconjugate of X(j) for these (i,j): {31, 3098}, {163, 9210}, {560, 7788}, {9247, 11331}
X(14387) = trilinear pole of line {850, 1637}
X(14387) = barycentric product X(523)*X(9211)
X(14387) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 3098}, {76, 7788}, {264, 11331}, {523, 9210}, {1637, 9411}, {9211, 99}

X(14388) =  ISOGONAL CONJUGATE OF X(11645)

Barycentrics    a^2 (2 a^6-2 a^4 b^2-2 a^2 b^4+2 b^6+a^4 c^2+b^4 c^2+a^2 c^4+b^2 c^4-4 c^6) (2 a^6+a^4 b^2+a^2 b^4-4 b^6-2 a^4 c^2+b^4 c^2-2 a^2 c^4+b^2 c^4+2 c^6) : :

X(14388) lies on the circumcircle, the cubic K330, and these lines: {3,11636}, {99,3534}, {107,5094}, {110,3098}, {112,574}, {376,6236}, {476,10989}, {1350,6233}, {2715,5104}, {7422,13530}, {7664,9100}

X(14388) = isogonal conjugate of X(11645)
X(14388) = circumcircle-antipode of X(11636)
X(14388) = trilinear pole of line {6, 9210}

X(14389) =  MIDPOINT OF X(5169) AND X(11003)

Barycentrics    2 a^6-3 a^4 b^2+b^6-3 a^4 c^2-2 a^2 b^2 c^2-b^4 c^2-b^2 c^4+c^6 : :
Barycentrics    a^2 - |OH|^2 + 3 R^2 : :
X(14389) = 3 X(2) + X(11004)

X(14389) lies on the cubic K502 and these lines: {2,6}, {4,6800}, {5,49}, {23,5480}, {52,6689}, {68,7569}, {83,2986}, {125,575}, {140,568}, {143,12363}, {146,7527}, {154,7394}, {182,858}, {184,3818}, {275,467}, {373,5972}, {378,4846}, {427,5012}, {458,5523}, {468,5640}, {511,7495}, {549,3581}, {569,1594}, {578,13160}, {1199,12359}, {1503,5169}, {1614,7403}, {1656,6090}, {1986,6699}, {2979,7499}, {3060,6676}, {3090,6193}, {3167,7539}, {3448,8550}, {3564,11422}, {3567,7542}, {3574,12225}, {3796,7391}, {4563,7769}, {5050,5094}, {5112,10796}, {5181,12039}, {5462,10018}, {5476,7426}, {5562,12242}, {5643,12596}, {6243,7568}, {6642,8907}, {6677,11451}, {7404,11441}, {7528,9707}, {7566,9833}, {7693,10192}, {7699,10297}, {7706,10295}, {9781,13383}, {9825,11449}, {10168,13857}, {11402,11442}, {12006,13416}, {12233,14118}, {12236,13363}, {13171,14130}, {13353,13371}

X(14389) = midpoint of X(5169) and X(11003)
X(14389) = X(7578)-complementary conjugate of X(2887)
X(14389) = crosspoint of X(2) and X(7578)
X(14389) = crossdifference of every pair of points on line {512, 2081}
X(14389) = crosssum of X(6) and X(566)
X(14389) = barycentric product X(69)*X(7576)
X(14389) = barycentric quotient X(7576)/X(4)
X(14389) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 6, 3580), (2, 323, 141), (2, 1994, 343), (2, 11427, 1993), (5, 567, 12022), (3589, 11064, 2), (5480, 13394, 23), (8836, 8838, 5)

X(14390) =  X(2)-CEVA CONJUGATE OF X(14379)

Barycentrics    f(A,B,C)( -f(A,B,C) + f(B,C,A) + f(C,A,B)) : : , where f(A,B,C) = (sin A)(cos^2 A)/(cos A - cos B cos C)
Barycentrics    a^4(a^8 - 2a^6(b^2 + c^2) + 10a^4b^2c^2 + 2a^2(b^2 + c^2)(b^4 - 4b^2c^2 + c^4) - (b^2 - c^2)^2(b^4 + 4b^2c^2 + c^4))/(3a^4 - b^4 - c^4 - 2a^2b^2 - 2a^2c^2 + 2b^2c^2) : :

The Yiu-Hutson conic is described at X(14379).

X(14390) lies on these lines: {6,64}, {800,11589}, {1249,13526}, {5065,14379}

X(14390) = isogonal conjugate of polar conjugate of X(39268)
X(14390) = X(2)-Ceva conjugate of X(14379)
X(14390) = X(i)-complementary conjugate of X(j) for these (i,j): {31, 14379}, {1096, 235}
X(14390) = center of Yiu-Hutson conic
X(14390) = barycentric product X(i)*X(j) for these {i,j}: {64, 11413}, {253, 1660}
X(14390) = barycentric quotient X(1660)/X(20)

X(14391) =  TRIPOLAR CENTROID OF X(5)

Barycentrics    (b^2-c^2) (-a^2+b^2+c^2) (-a^2 b^2+b^4-a^2 c^2-2 b^2 c^2+c^4) (-2 a^4+a^2 b^2+b^4+a^2 c^2-2 b^2 c^2+c^4) : :
X(14391) = 3 X(1636) - 4 X(14345) = 3 X(1637) - 2 X(14345)

Tripolar centroids are discussed in the preamble to X(1635).

X(14392) lies on these lines: {450, 2451}, {1636, 1637}, {1640, 9517}, {2081, 2600}, {3448, 13527}

X(14391) = reflection of X(1636) in X(1637)
X(14391) = crosspoint of X(i) and X(j) for these (i,j): {265, 648}
X(14391) = crossdifference of every pair of points on line {54, 74, 185, 933, 2914, 3520, 7722, 10628, 11430, 11587, 12227, 13198, 13293}
X(14391) = crosssum of X(i) and X(j) for these (i,j): {186, 647}
X(14391) = X(i)-isoconjugate of X(j) for these (i,j): {933, 2349}, {1304, 2167}
X(14391) = barycentric product X(i)*X(j) for these {i,j}: {5, 9033}, {30, 6368}, {311, 9409}, {324, 1636}, {343, 1637}, {523, 1568}, {2631, 14213}, {11064, 12077}, {13157, 14345}
X(14391) = barycentric quotient X(i)/X(j) for these {i,j}: {51, 1304}, {1495, 933}, {1568, 99}, {1636, 97}, {1637, 275}, {2631, 2167}, {6368, 1494}, {9033, 95}, {9409, 54}

X(14392) =  TRIPOLAR CENTROID OF X(9)

Barycentrics    a*(a - b - c)^2*(b - c)*(2*a^2 - a*b - b^2 - a*c + 2*b*c - c^2) : :
X(14392) = 2 X(650) + X(4105) = 4 X(650) - X(6608) = 2 X(4105) + X(6608)

X(14392) lies on these lines: {100, 109}, {650, 663}, {926, 1635}, {1637, 1962}, {1638, 6174}, {2310, 3119}, {4893, 6182}

X(14392) = reflection of X(1635) in X(11124)
X(14492) = isotomic conjugate of X(37671)
X(14392) = crossdifference of every pair of points on line {57, 934, 2170, 2291}
X(14392) = crosssum of X(i) and X(j) for these (i,j): {513, 6610}
X(14392) = {X(650),X(4105)}-harmonic conjugate of X(6608)
X(14392) = centroid of (degenerate) side-triangle of ABC and Gemini triangle 36
X(14392) = X(i)-isoconjugate of X(j) for these (i,j): {658, 2291}, {934, 1156}, {1121, 1461}, {4626, 4845}
X(14392) = barycentric product X(i)*X(j) for these {i,j}: {9, 6366}, {200, 1638}, {312, 6139}, {522, 6603}, {527, 3900}, {650, 6745}, {1055, 4397}, {1155, 3239}, {1323, 4130}, {4163, 6610}
X(14392) = barycentric quotient X(i)/X(j) for these {i,j}: {527, 4569}, {657, 1156}, {1055, 934}, {1155, 658}, {1638, 1088}, {3900, 1121}, {6139, 57}, {6366, 85}, {6603, 664}, {6610, 4626}, {6745, 4554}, {8641, 2291}

X(14393) =  TRIPOLAR CENTROID OF X(11)

Barycentrics    (b-c)^3 (-a+b+c) (-a b+b^2-a c+c^2) (-2 a^3+2 a^2 b-a b^2+b^3+2 a^2 c-b^2 c-a c^2-b c^2+c^3) : :

X(14393) lies on these lines: {294, 885}, {903, 918}, {1642, 1643}

X(14393) = crossdifference of every pair of points on line {59, 840, 1362}
X(14393) = barycentric product X(528)*X(3717)*X(6545)

X(14394) =  TRIPOLAR CENTROID OF X(12)

Barycentrics    (b-c) (b+c)^2 (a b+b^2+a c+c^2) (2 a^4-a^2 b^2-b^4+4 a^2 b c-2 a b^2 c-a^2 c^2-2 a b c^2+2 b^2 c^2-c^4) : :

X(14394) lies on this line: {2610, 4024}
X(14394) = barycentric product {529,4024,4357}, {523,529,1211}

X(14395) =  TRIPOLAR CENTROID OF X(21)

Barycentrics    a*(a - b - c)*(b - c)*(a^2 - b^2 - c^2)*(2*a^4 - a^2*b^2 - b^4 - a^2*c^2 + 2*b^2*c^2 - c^4) : :

X(14395) lies on these lines: {521, 650}, {652, 2523}, {1635, 8677}, {1636, 1637}, {7004, 7117}

X(14395) = cevapoint of X(i) and X(j) for these (i,j): {1636, 2631}
X(14395) = crosspoint of X(i) and X(j) for these (i,j): {651, 1807}
X(14395) = crossdifference of every pair of points on line {65, 74, 108, 2778, 5494, 6198, 7414, 10118}
X(14395) = crosssum of X(i) and X(j) for these (i,j): {650, 1870}
X(14395) = centroid of (degenerate) cross-triangle of anticevian triangles of X(1) and X(3)
X(14395) = X(i)-isoconjugate of X(j) for these (i,j): {74, 653}, {108, 2349}, {226, 1304}, {664, 8749}
X(14395) = barycentric product X(i)*X(j) for these {i,j}: {21, 9033}, {30, 521}, {78, 11125}, {314, 9409}, {333, 2631}, {650, 11064}, {652, 14206}, {905, 7359}, {1637, 1812}, {1946, 3260}, {2173, 6332}, {3284, 4391}
X(14395) = barycentric quotient X(i)/X(j) for these {i,j}: {521, 1494}, {652, 2349}, {1495, 108}, {1636, 1214}, {1946, 74}, {2173, 653}, {2194, 1304}, {2631, 226}, {3063, 8749}, {3284, 651}, {6357, 13149}, {7359, 6335}, {9033, 1441}, {9409, 65}, {11064, 4554}, {11125, 273}

X(14396) =  TRIPOLAR CENTROID OF X(22)

Barycentrics    a^2*(b - c)*(b + c)*(a^2 - b^2 - c^2)*(a^4 - b^4 - c^4)*(2*a^4 - a^2*b^2 - b^4 - a^2*c^2 + 2*b^2*c^2 - c^4) : :

X(14396) lies on these lines: {520, 9210}, {1636, 1637}, {2485, 8673}

X(14396) = crosspoint of X(2420) and X(11064)
X(14396) = crossdifference of every pair of points on line {66, 74}
X(14396) = crosssum of X(2394) and X(8749)
X(14396) = isoconjugate of X(1289) and X(2349)
X(14396) = barycentric product X(i)*X(j) for these {i,j}: {22, 9033}, {30, 8673}, {127, 2420}, {315, 9409}, {1760, 2631}, {2485, 11064}
X(14396) = barycentric quotient X(i)/X(j) for these {i,j}: {206, 1304}, {1495, 1289}, {8673, 1494}, {9409, 66}

X(14397) =  TRIPOLAR CENTROID OF X(24)

Barycentrics    a^2*(b - c)*(b + c)*(2*a^4 - a^2*b^2 - b^4 - a^2*c^2 + 2*b^2*c^2 - c^4)*(a^4 - 2*a^2*b^2 + b^4 - 2*a^2*c^2 + c^4) : :

X(14397) lies on these lines: {6, 686}, {51, 351}, {647, 657}, {924, 6753}, {1636, 1637}, {10982, 11615}

X(14397) = X(1289)-isoconjugate of X(2349)
X(14397) = crosspoint of X(i) and X(j) for these (i,j): {1990, 2420}
X(14397) = crossdifference of every pair of points on line {20, 68}
X(14397) = crosssum of X(2) and X(6334)
X(14397) = barycentric product X(i)*X(j) for these {i,j}: {24, 9033}, {30, 924}, {317, 9409}, {1495, 6563}, {1636, 11547}, {1637, 1993}, {1748, 2631}, {6753, 11064}
X(14397) = barycentric quotient X(i)/X(j) for these {i,j}: {924, 1494}, {1495, 925}, {1637, 5392}, {9409, 68}

X(14398) =  TRIPOLAR CENTROID OF X(25)

Trilinears    (sin A tan A)(tan B - tan C)(tan B + tan C - 2 tan A) : :
Barycentrics    a^2 S (SB - SC) (S^2 - 3 SB SC) : :
Barycentrics    a^2*(b - c)*(b + c)*(2*a^4 - a^2*b^2 - b^4 - a^2*c^2 + 2*b^2*c^2 - c^4) : :
X(14398) = X(2451) + 2 X(2485) = X(6) + 2 X(2492) = 2 X(2489) + X(3049) = 4 X(2492) - X(3569) = 2 X(6) + X(3569) = 2 X(2030) + X(8430) = X(351) + 2 X(9171) = X(9135) + 2 X(9178) = X(9135) - 4 X(9188) = X(9178) + 2 X(9188)

X(14398) lies on these lines: {2, 9035}, {6, 526}, {351, 865}, {512, 1692}, {684, 10567}, {804, 6034}, {1636, 1637}, {1643, 6089}, {1976, 2422}, {2030, 8430}, {2433, 8749}, {2451, 2485}, {2508, 2869}, {2780, 5622}, {9408, 9409}

X(14398) = midpoint of X(2451) and X(9210)
X(14398) = reflection of X(9210) in X(2485)
X(14398) = X(i)-Ceva conjugate of X(j) for these (i,j): {1637, 9409}, {2420, 1495}, {2433, 512}, {11060, 3124}
X(14398) = X(i)-isoconjugate of X(j) for these (i,j): {74, 799}, {99, 2349}, {304, 1304}, {662, 1494}, {670, 2159}
X(14398) = X(i)-Hirst inverse of X(j) for these (i,j): {865, 888}, {3124, 6041}
X(14398) = crosspoint of X(i) and X(j) for these (i,j): {512, 2433}, {1495, 2420}
X(14398) = crossdifference of every pair of points on line {69, 74, 99, 376, 542, 616, 617, 1272, 1494, 3098, 6148, 7811, 9862, 11006, 11128, 11129, 12317, 12383, 13169, 13210}
X(14398) = crosssum of X(i) and X(j) for these (i,j): {2, 3268}, {99, 2407}, {1494, 2394}, {3265, 11064}, {4558, 10411}
X(14398) = barycentric product X(i)*X(j) for these {i,j}: {4, 9409}, {6, 1637}, {19, 2631}, {25, 9033}, {30, 512}, {42, 11125}, {115, 2420}, {351, 9214}, {393, 1636}, {523, 1495}, {647, 1990}, {661, 2173}, {669, 3260}, {691, 2682}, {798, 14206}, {810, 1784}, {850, 9407}, {1577, 9406}, {2394, 9408}, {2407, 3124}, {2433, 3163}, {2489, 11064}, {2501, 3284}, {3471, 6140}, {3709, 6357}, {5642, 9178}, {5664, 11060}, {7180, 7359}, {14254, 14270}
X(14398) = barycentric quotient X(i)/X(j) for these {i,j}: {30, 670}, {512, 1494}, {669, 74}, {798, 2349}, {1084, 2433}, {1495, 99}, {1636, 3926}, {1637, 76}, {1924, 2159}, {1974, 1304}, {1990, 6331}, {2173, 799}, {2420, 4590}, {2631, 304}, {3124, 2394}, {3260, 4609}, {3284, 4563}, {9033, 305}, {9406, 662}, {9407, 110}, {9408, 2407}, {9409, 69}, {9411, 7788}, {11125, 310}, {14206, 4602}
X(14398) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6, 2492, 3569), (2491, 6041, 351), (9178, 9188, 9135)

X(14399) =  TRIPOLAR CENTROID OF X(28)

Barycentrics    a*(b - c)*(2*a^4 - a^2*b^2 - b^4 - a^2*c^2 + 2*b^2*c^2 - c^4) : :

X(14399) lies on these lines: {244, 665}, {513, 1430}, {1636, 1637}

X(14399) = X(i)-isoconjugate of X(j) for these (i,j): {74, 190}, {100, 2349}, {101, 1494}, {306, 1304}, {668, 2159}, {2394, 4570}, {2433, 4600}, {4561, 8749}
X(14399) = X(866)-Hirst inverse of X(891)
X(14399) = crossdifference of every pair of points on line {72, 74}
X(14399) = barycentric product X(i)*X(j) for these {i,j}: {1, 11125}, {27, 2631}, {28, 9033}, {30, 513}, {81, 1637}, {286, 9409}, {514, 2173}, {649, 14206}, {650, 6357}, {667, 3260}, {693, 1495}, {905, 1990}, {1459, 1784}, {2407, 3125}, {3261, 9406}, {3669, 7359}, {6591, 11064}
X(14399) = barycentric quotient X(i)/X(j) for these {i,j}: {30, 668}, {513, 1494}, {649, 2349}, {667, 74}, {1495, 100}, {1636, 3998}, {1637, 321}, {1919, 2159}, {1990, 6335}, {2173, 190}, {2203, 1304}, {2407, 4601}, {2420, 4567}, {2631, 306}, {3121, 2433}, {3125, 2394}, {3260, 6386}, {3284, 1332}, {6357, 4554}, {7359, 646}, {9406, 101}, {9407, 692}, {9409, 72}, {11125, 75}, {14206, 1978}

X(14400) =  TRIPOLAR CENTROID OF X(29)

Barycentrics    (a - b - c)*(b - c)*(2*a^4 - a^2*b^2 - b^4 - a^2*c^2 + 2*b^2*c^2 - c^4) : :
X(14400) = X(652) + 2 X(3064) = X(652) - 4 X(14331) = X(3064) + 2 X(14331)

X(14400) lies on these lines: {9, 14224}, {11, 1146}, {243, 522}, {650, 4802}, {654, 4984}, {1636, 1637}, {4024, 9404}, {4120, 8674}, {4944, 14298}

X(14400) = cevapoint of X(i) and X(j) for these (i,j): {1637, 2631}
X(14400) = crosspoint of X(i) and X(j) for these (i,j): {80, 653}
X(14400) = crossdifference of every pair of points on line {35, 73, 74, 109, 2779, 4337, 6126, 7421, 10088}
X(14400) = crosssum of X(i) and X(j) for these (i,j): {36, 652}
X(14400) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3064, 14331, 652)
X(14400) = centroid of (degenerate) cross-triangle of anticevian triangles of X(1) and X(4)
X(14400) = X(i)-isoconjugate of X(j) for these (i,j): {74, 651}, {109, 2349}, {664, 2159}, {1214, 1304}, {1415, 1494}, {6516, 8749}
X(14400) = barycentric product X(i)*X(j) for these {i,j}: {8, 11125}, {29, 9033}, {30, 522}, {333, 1637}, {514, 7359}, {521, 1784}, {650, 14206}, {663, 3260}, {1990, 6332}, {2173, 4391}, {3064, 11064}, {3239, 6357}, {6357, 8834}
X(14400) = barycentric quotient X(i)/X(j) for these {i,j}: {30, 664}, {522, 1494}, {650, 2349}, {663, 74}, {1495, 109}, {1637, 226}, {1990, 653}, {2173, 651}, {2299, 1304}, {2407, 4620}, {2631, 1214}, {3063, 2159}, {3260, 4572}, {3284, 1813}, {6357, 658}, {7359, 190}, {9033, 307}, {9406, 1415}, {9409, 73}, {11125, 7}, {14206, 4554}

X(14401) =  TRIPOLAR CENTROID OF X(30)

Barycentrics    (b - c)*(b + c)*(-a^2 + b^2 + c^2)*(-2*a^4 + a^2*b^2 + b^4 + a^2*c^2 - 2*b^2*c^2 + c^4)^2 : :

X(14401) lies on these on lines: {2, 525}, {6, 2430}, {113, 1560}, {216, 647}, {373, 520}, {1249, 2501}, {1636, 1637}, {2420, 4240}

X(14401) = isogonal conjugate of X(34568)
X(14401) = complement of X(34767)
X(14401) = X(i)-complementary conjugate of X(j) for these (i,j): {31, 1650}, {2173, 127}, {4240, 2887}
X(14401) = X(i)-Ceva conjugate of X(j) for these (i,j): {2, 1650}, {525, 9033}, {648, 30}, {4240, 3081}
X(14401) = crosspoint of X(i) and X(j) for these (i,j): {2, 4240}, {30, 648}, {525, 9033}
X(14401) = crossdifference of every pair of points on line {74, 186}
X(14401) = crosssum of X(i) and X(j) for these (i,j): {74, 647}, {112, 1304}
X(14401) = X(1304)-isoconjugate of X(2349)
X(14401) = barycentric product X(i)*X(j) for these {i,j}: {30, 9033}, {125, 3233}, {525, 3163}, {656, 1099}, {1553, 14220}, {1637, 11064}, {1650, 4240}, {2631, 14206}, {3260, 9409}, {3267, 9408}
X(14401) = barycentric quotient X(i)/X(j) for these {i,j}: {1099, 811}, {1495, 1304}, {2631, 2349}, {3081, 4240}, {3163, 648}, {9033, 1494}, {9408, 112}, {9409, 74}

X(14402) =  TRIPOLAR CENTROID OF X(31)

Barycentrics    a^3*(b - c)*(b^2 + b*c + c^2)*(2*a^3 - b^3 - c^3) : :

X(14402) lies on these on lines: {2, 794}, {667, 788}

X(14402) = X(870)-isoconjugate of X(5386)
X(14402) = crossdifference of every pair of points on line {75, 753}
X(14402) = barycentric product X(i)*X(j) for these {i,j}: {752, 788}, {869, 4809}, {1491, 8626}, {2243, 3250}
X(14402) = barycentric quotient X(i)/X(j) for these {i,j}: {4809, 871}, {8626, 789}, {8630, 753}

X(14403) =  TRIPOLAR CENTROID OF X(32)

Barycentrics    a^4*(b - c)*(b + c)*(b^2 + c^2)*(2*a^4 - b^4 - c^4) : :

X(14403) lies on these on lines: {2, 782}, {351, 11205}, {669, 688}, {732, 11176}, {804, 10191}, {9019, 9188}

X(14403) = midpoint of X(i) and X(j) for these {i,j}: {351, 11205}
X(14403) = crossdifference of every pair of points on line {76, 689}
X(14403) = barycentric product X(i)*X(j) for these {i,j}: {688, 754}, {2084, 2244}, {3005, 8627}
X(14403) = barycentric quotient X(i)/X(j) for these {i,j}: {8627, 689}, {9494, 755}

X(14404) =  TRIPOLAR CENTROID OF X(37)

Barycentrics    a^2*(b - c)*(b + c)*(a*b + a*c - 2*b*c) : :
X(14404) = 2 X(4507) + X(4813)

X(14404) lies on these on lines: {6, 5040}, {42, 4455}, {351, 865}, {512, 661}, {650, 9010}, {788, 4893}, {890, 3768}, {891, 4728}, {1491, 9002}, {1635, 6373}, {2254, 3030}, {2703, 5380}, {3952, 4010}, {4083, 4776}, {4120, 4155}, {4507, 4813}

X(14404) = reflection of X(8027) in X(1635)
X(14404) = X(899)-Ceva conjugate of X(1646)
X(14404) = crossdifference of every pair of points on line {81, 99}
X(14404) = crosssum of X(898) and X(4607)
X(14404) = X(i)-isoconjugate of X(j) for these (i,j): {58, 889}, {81, 4607}, {86, 898}, {662, 3227}, {739, 799}, {1019, 5381}
X(14404) = barycentric product X(i)*X(j) for these {i,j}: {10, 3768}, {37, 891}, {42, 4728}, {65, 4526}, {321, 890}, {512, 536}, {523, 3230}, {649, 3994}, {661, 899}, {798, 6381}, {1646, 3952}, {4009, 7180}
X(14404) = barycentric quotient X(i)/X(j) for these {i,j}: {37, 889}, {42, 4607}, {213, 898}, {512, 3227}, {536, 670}, {669, 739}, {890, 81}, {891, 274}, {899, 799}, {1646, 7192}, {3230, 99}, {3768, 86}, {3994, 1978}, {4526, 314}, {4557, 5381}, {4728, 310}, {6381, 4602}

X(14405) =  TRIPOLAR CENTROID OF X(38)

Barycentrics    a*(b - c)*(a^2 - b*c)*(b^2 + c^2)*(a^2*b - 2*a*b^2 + a^2*c + b^2*c - 2*a*c^2 + b*c^2) : :

X(14405) lies on this line: {2084, 2530}

X(14405) = crossdifference of every pair of points on line {82, 2382}

X(14406) =  TRIPOLAR CENTROID OF X(39)

Barycentrics    a^4*(b - c)*(b + c)*(b^2 + c^2)*(a^2*b^2 + a^2*c^2 - 2*b^2*c^2) : :

X(14406) lies on these on lines: {351, 9429}, {512, 9147}, {688, 3005}, {888, 6786}, {3221, 5996}, {5113, 9998}

X(14406) = X(3231)-Ceva conjugate of X(1645)
X(14406) = crosspoint of X(i) and X(j) for these (i,j): {887, 888}
X(14406) = crossdifference of every pair of points on line {83, 689, 729, 13518}
X(14406) = crosssum of X(i) and X(j) for these (i,j): {886, 9150}
X(14406) = X(i)-isoconjugate of X(j) for these (i,j): {82, 886}, {3112, 9150}, {3228, 4593}
X(14406) = barycentric product X(i)*X(j) for these {i,j}: {39, 888}, {141, 887}, {538, 688}, {1645, 4576}, {2084, 2234}, {3005, 3231}, {3051, 9148}
X(14406) = barycentric quotient X(i)/X(j) for these {i,j}: {39, 886}, {688, 3228}, {887, 83}, {888, 308}, {3051, 9150}, {3231, 689}, {9494, 729}

X(14407) =  TRIPOLAR CENTROID OF X(42)

Barycentrics    a^2*(2*a - b - c)*(b - c)*(b + c) : :
X(14407) = X(798) + 2 X(3709) = 2 X(665) + X(3768) = 4 X(3709) - X(4079) = 2 X(798) + X(4079) = 5 X(4079) - 2 X(4826) = 10 X(3709) - X(4826) = 5 X(798) + X(4826) = 5 X(798) - 2 X(4832) = 5 X(3709) + X(4832) = X(4826) + 2 X(4832) = 5 X(4079) + 4 X(4832)

X(14407) lies on these on lines: {6, 5029}, {37, 4145}, {351, 865}, {512, 798}, {573, 2776}, {649, 4491}, {657, 1919}, {665, 3768}, {900, 1635}, {1017, 1960}, {2054, 9178}, {6586, 9002}

X(14407) = isogonal conjugate of X(4615)
X(14407) = X(1635)-Ceva conjugate of X(4730)
X(14407) = crosspoint of X(i) and X(j) for these (i,j): {101, 6187}, {1635, 1960}
X(14407) = crossdifference of every pair of points on line {86, 99, 106, 551, 903, 2796, 3663, 4234, 4653, 7321}
X(14407) = crosssum of X(i) and X(j) for these (i,j): {320, 514}, {903, 4049}, {3257, 4555}
X(14407) = barycentric product X(i)*X(j) for these {i,j}: {1, 4730}, {6, 4120}, {10, 1960}, {37, 1635}, {42, 900}, {44, 661}, {65, 4895}, {213, 3762}, {512, 519}, {523, 902}, {647, 8756}, {649, 3943}, {667, 3992}, {669, 3264}, {798, 4358}, {850, 9459}, {875, 4783}, {1015, 4169}, {1017, 4049}, {1018, 2087}, {1023, 3125}, {1042, 4528}, {1319, 4041}, {1400, 1639}, {1402, 4768}, {1404, 3700}, {1577, 2251}, {1647, 4557}, {2325, 7180}, {2489, 3977}, {3251, 4674}, {3285, 4024}, {3689, 4017}, {3709, 3911}, {4530, 4559}
X(14407) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 4634}, {6, 4615}, {31, 4622}, {32, 4591}, {42, 4555}, {44, 799}, {213, 3257}, {512, 903}, {519, 670}, {669, 106}, {798, 88}, {900, 310}, {902, 99}, {1023, 4601}, {1319, 4625}, {1334, 4582}, {1404, 4573}, {1635, 274}, {1918, 901}, {1924, 9456}, {1960, 86}, {2087, 7199}, {2251, 662}, {2489, 6336}, {3049, 1797}, {3121, 1022}, {3122, 6548}, {3124, 4049}, {3264, 4609}, {3285, 4610}, {3689, 7257}, {3709, 4997}, {3762, 6385}, {3943, 1978}, {3992, 6386}, {4079, 4080}, {4120, 76}, {4358, 4602}, {4730, 75}, {4895, 314}, {8034, 6549}, {8756, 6331}, {9459, 110}
X(14407) = X(i)-isoconjugate of X(j) for these (i,j): {1, 4615}, {2, 4622}, {6, 4634}, {75, 4591}, {81, 4555}, {86, 3257}, {88, 99}, {106, 799}, {274, 901}, {662, 903}, {670, 9456}, {811, 1797}, {1014, 4582}, {1022, 4600}, {1320, 4573}, {1414, 4997}, {2316, 4625}, {4567, 6548}, {4592, 6336}, {4610, 4674}, {5376, 7192}, {7199, 9268}

X(14408) =  TRIPOLAR CENTROID OF X(43)

Barycentrics    a*(2*a - b - c)*(b - c)*(a*b + a*c - b*c) : :
X(14408) = 2 X(37) + X(3768)

X(14408) lies on these on lines: {9, 8632}, {37, 3768}, {45, 4491}, {661, 1769}, {888, 1962}, {900, 1635}, {3123, 6377}, {3251, 9461}, {4502, 6005}

X(14408) = X(1960)-Ceva conjugate of X(1635)
X(14408) = crossdifference of every pair of points on line {87, 106, 932, 993}
X(14408) = barycentric product X(i)*X(j) for these {i,j}: {43, 900}, {44, 3835}, {192, 1635}, {519, 4083}, {1319, 4147}, {1403, 4768}, {1423, 1639}, {1960, 6376}, {2087, 4595}, {2176, 3762}, {3212, 4895}, {3264, 8640} X(14408) = barycentric quotient X(i)/X(j) for these {i,j}: {43, 4555}, {44, 4598}, {900, 6384}, {902, 932}, {1023, 5383}, {1635, 330}, {1960, 87}, {2176, 3257}, {2209, 901}, {3123, 6548}, {3208, 4582}, {3762, 6383}, {4083, 903}, {4895, 7155}, {6377, 1022}, {8640, 106}
X(14408) = X(i)-isoconjugate of X(j) for these (i,j): {87, 3257}, {88, 932}, {106, 4598}, {330, 901}, {2162, 4555}

X(14409) =  TRIPOLAR CENTROID OF X(44)

Barycentrics    a*(a - 2*b - 2*c)*(2*a - b - c)*(b - c)*(4*a^2 - a*b - 2*b^2 - a*c + 2*b*c - 2*c^2) : :

X(14409) lies on these on lines: {678, 1635}, {2177, 4825}, {4448, 4664}

X(14410) =  TRIPOLAR CENTROID OF X(45)

Barycentrics    a*(a - 2*b - 2*c)*(2*a - b - c)*(b - c)*(2*a^2 - 2*a*b - b^2 - 2*a*c + 4*b*c - c^2) : :

X(14410) lies on these on lines: {1635, 8661}, {4770, 4775}

X(14410) = crossdifference of every pair of points on line {89, 2384}
X(14410) = barycentric product X(i)*X(j) for these {i,j}: {1644, 4893}

X(14411) =  TRIPOLAR CENTROID OF X(55)

Barycentrics    a^2*(a - b - c)*(b - c)*(a*b - b^2 + a*c - c^2)*(2*a^3 - 2*a^2*b + a*b^2 - b^3 - 2*a^2*c + b^2*c + a*c^2 + b*c^2 - c^3) : :

X(14411) lies on these on lines: {657, 663}, {665, 672}, {1642, 1643}, {2284, 5548}

X(14411) = crossdifference of every pair of points on line {7, 840}
X(14411) = barycentric product X(i)*X(j) for these {i,j}: {528, 926}, {650, 1642}, {1643, 3693}
X(14411) = barycentric quotient X(i)/X(j) for these {i,j}: {1642, 4554}, {8638, 840}

X(14412) =  TRIPOLAR CENTROID OF X(56)

Barycentrics    a^2*(b - c)*(a*b + b^2 + a*c + c^2)*(2*a^4 - a^2*b^2 - b^4 + 4*a^2*b*c - 2*a*b^2*c - a^2*c^2 - 2*a*b*c^2 + 2*b^2*c^2 - c^4) : :

X(14412) lies on this on line: {649, 854}

X(14412) = crossdifference of every pair of points on line {8, 8707}
X(14412) = barycentric product X(i)*X(j) for these {i,j}: {529, 6371}

X(14413) =  TRIPOLAR CENTROID OF X(57)

Barycentrics    a*(b - c)*(2*a^2 - a*b - b^2 - a*c + 2*b*c - c^2) : :
X(14413) = 2 X(1)+X(2254) = X(1)+2 X(3960) = 4 X(1)-X(4895) = X(2254)-4 X(3960) = 2 X(2254)+X(4895) = 8 X(3960)+X(4895) = X(663)+2 X(3669) = 2 X(1459)+X(4017) = 4 X(6129)-X(6615) = X(661)+2 X(4378) = X(764)+2 X(1960) = 4 X(905)-X(4041) = 2 X(905)+X(4449) = X(4041)+2 X(4449) = 4 X(1125)-X(3762) = 5 X(3616)-2 X(3716) = 2 X(3837)+X(4922) = X(3904)+2 X(4458) = X(4474)-4 X(4885)

X(14413) lies on these on lines:
{1, 2254}, {56, 8648}, {105, 106}, {109, 934}, {244, 665}, {513, 663}, {661, 4378}, {690, 1962}, {764, 1960}, {905, 4041}, {1125, 3762}, {1411, 2424}, {1636, 9391}, {1638, 6174}, {1946, 7216}, {2785, 4453}, {2787, 4728}, {2814, 3576}, {3616, 3716}, {3837, 4922}, {3904, 4458}, {4474, 4885}

X(14413) = X(i)-Hirst inverse of X(j) for these (i,j): {244, 1643}, {1635, 3675}
X(14413) = crosspoint of X(i) and X(j) for these (i,j): {1, 1308}
X(14413) = crossdifference of every pair of points on line {9, 100, 1005, 1156, 2246, 2291, 3119, 5528, 6594, 7676}
X(14413) = crosssum of X(i) and X(j) for these (i,j): {1, 3887}, {3900, 6603}
X(14413) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 2254, 4895), (1, 3960, 2254), (665, 1643, 1635), (905, 4449, 4041)
X(14413) = X(21)-beth conjugate of X(3887)
X(14413) = centroid of (degenerate) side-triangle of ABC and Gemini triangle 35
X(14413) = centroid of (degenerate) side-triangle of ABC and Gemini triangle 40
X(14413) = X(i)-isoconjugate of X(j) for these (i,j): {100, 1156}, {101, 1121}, {190, 2291}, {664, 4845}
X(14413) = barycentric product X(i)*X(j) for these {i,j}: {1, 1638}, {57, 6366}, {85, 6139}, {513, 527}, {514, 1155}, {522, 6610}, {650, 1323}, {693, 1055}, {1022, 6174}, {3669, 6745}, {3676, 6603}, {6510, 7649} X(14413) = barycentric quotient X(i)/X(j) for these {i,j}: {513, 1121}, {527, 668}, {649, 1156}, {667, 2291}, {1055, 100}, {1155, 190}, {1323, 4554}, {1638, 75}, {3063, 4845}, {6139, 9}, {6366, 312}, {6510, 4561}, {6603, 3699}, {6610, 664}, {6745, 646}

X(14414) =  TRIPOLAR CENTROID OF X(63)

Barycentrics    a*(a - b - c)*(b - c)*(a^2 - b^2 - c^2)*(2*a^2 - a*b - b^2 - a*c + 2*b*c - c^2) : :

X(14414) lies on these on lines: {521, 656}, {891, 11124}, {1331, 6516}, {1638, 6174}, {2804, 11125}, {2812, 3576}, {7004, 7117}

X(14414) = crossdifference of every pair of points on line {19, 108, 2291}
X(14414) = centroid of (degenerate) side-triangle of ABC and hexyl triangle
X(14414) = X(i)-isoconjugate of X(j) for these (i,j): {108, 1156}, {653, 2291}
X(14414) = barycentric product X(i)*X(j) for these {i,j}: {63, 6366}, {78, 1638}, {304, 6139}, {521, 527}, {522, 6510}, {905, 6745}, {1155, 6332}, {4025, 6603}
X(14414) = barycentric quotient X(i)/X(j) for these {i,j}: {521, 1121}, {652, 1156}, {1055, 108}, {1155, 653}, {1323, 13149}, {1638, 273}, {1946, 2291}, {6139, 19}, {6366, 92}, {6510, 664}, {6603, 1897}, {6745, 6335}

X(14415) =  TRIPOLAR CENTROID OF X(65)

Barycentrics    a^2*(b - c)*(b + c)*(a^2 + a*b + a*c + 2*b*c)*(3*a^2*b - 3*b^3 + 3*a^2*c - 2*a*b*c + b^2*c + b*c^2 - 3*c^3) : :

X(14415) lies on this line: {647, 661}

X(14415) = crossdifference of every pair of points on line {21, 931}

X(14416) =  TRIPOLAR CENTROID OF X(66)

Barycentrics    (b - c)*(b + c)*(b^2 + c^2)*(-2*a^8 + a^4*b^4 + b^8 + a^4*c^4 - 2*b^4*c^4 + c^8) : :

X(14416) lies on these on lines: {647, 826}, {1636, 11205}

X(14416) = crossdifference of every pair of points on line {22, 827}

X(14417) =  TRIPOLAR CENTROID OF X(69)

Barycentrics    (b - c)*(b + c)*(-2*a^2 + b^2 + c^2)*(-a^2 + b^2 + c^2) : :
Barycentrics    cot A (cot B - cot C)(cot B + cot C - 2 cot A) : :
Barycentrics    (tan B - tan C)(cot B + cot C - 2 cot A) : :
X(14417) = X(351) - 3 X(1649) = 2 X(647) + X(2525) = X(2525) - 4 X(3265) = X(647) + 2 X(3265) = X(1637) + 2 X(3268) = X(6334) + 2 X(8552) = 2 X(351) - 3 X(9125) = X(9131) - 3 X(9168) = X(9131) + 3 X(9191) = 3 X(1637) - 2 X(9979) = 3 X(3268) + X(9979) = 2 X(6563) + X(12077)

X(14417) lies on these on lines:
{2, 1637}, {4, 9529}, {122, 125}, {126, 1560}, {351, 690}, {441, 525}, {523, 7625}, {599, 9003}, {686, 13302}, {804, 10190}, {1638, 6370}, {2793, 6054}, {2848, 10714}, {3005, 7927}, {3566, 8644}, {3806, 7950}, {4558, 4563}, {4728, 6089}, {5108, 8429}, {6088, 14279}, {6563, 12077}, {9035, 10567}, {9148, 11123}, {9479, 11176}

X(14417) = midpoint of X(i) and X(j) for these {i,j}: {2, 3268}, {9148, 11123}, {9168, 9191}, {9204, 9205}
X(14417) = reflection of X(i) in X(j) for these {i,j}: {1637, 2}, {9125, 1649}
X(14417) = isogonal conjugate of polar conjugate of X(35522)
X(14417) = complement X(9979)
X(14417) = X(i)-complementary conjugate of X(j) for these (i,j): {163, 6593}, {2157, 125}, {3455, 8287}
X(14417) = X(i)-Ceva conjugate of X(j) for these (i,j): {4235, 524}, {5467, 7813}, {5468, 3292}
X(14417) = crosspoint of X(i) and X(j) for these (i,j): {524, 4235}, {3266, 5468}
X(14417) = crossdifference of every pair of points on line {25, 111}
X(14417) = crosssum of X(i) and X(j) for these (i,j): {3, 13303}, {6, 2492}, {110, 11634}, {111, 10097}, {2501, 5523}, {8744, 10561}
X(14417) = X(i)-isoconjugate of X(j) for these (i,j): {19, 691}, {92,32729}, {111, 162}, {112, 897}, {648, 923}, {662, 8753}, {892, 1973}, {1474, 5380}
X(14417) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (647, 3265, 2525)
X(14417) = barycentric product X(i)*X(j) for these {i,j}: {69, 690}, {125, 5468}, {187, 3267}, {304, 2642}, {305, 351}, {306, 4750}, {339, 5467}, {468, 3265}, {523, 6390}, {524, 525}, {647, 3266}, {656, 14210}, {850, 3292}, {896, 14208}, {1648, 4563}, {3926, 14273}, {4025, 4062}, {4064, 6629}, {4580, 7813}, {5967, 6333}
X(14417) = barycentric quotient X(i)/X(j) for these {i,j}: {3, 691}, {69, 892}, {72, 5380}, {125, 5466}, {187, 112}, {351, 25}, {468, 107}, {512, 8753}, {520, 895}, {524, 648}, {525, 671}, {647, 111}, {656, 897}, {684, 5968}, {690, 4}, {810, 923}, {879, 9154}, {896, 162}, {1648, 2501}, {1649, 468}, {2482, 4235}, {2642, 19}, {3266, 6331}, {3269, 10097}, {3292, 110}, {4062, 1897}, {4750, 27}, {5467, 250}, {5642, 4240}, {5967, 685}, {6390, 99}, {9033, 9214}, {9125, 4232}, {9155, 4230}, {9177, 7482}, {9204, 470}, {9205, 471}, {9517, 14246}, {9717, 1304}, {10097, 10630}, {11183, 419}, {14210, 811}, {14273, 393}, {14357, 935}

X(14418) =  TRIPOLAR CENTROID OF X(78)

Barycentrics    a*(a - b - c)*(2*a - b - c)*(b - c)*(a^2 - b^2 - c^2) : :
X(14418) = 2 X(652) + X(8611)

X(14418) lies on these on lines: {521, 652}, {650, 6615}, {900, 1635}, {4571, 4587}, {7004, 7117}

X(14418) = crosspoint of X(i) and X(j) for these (i,j): {1023, 2325}
X(14418) = crossdifference of every pair of points on line {34, 106, 108, 1420, 1421, 2840, 4222}
X(14418) = X(i)-isoconjugate of X(j) for these (i,j): {34, 3257}, {88, 108}, {106, 653}, {109, 6336}, {225, 4591}, {278, 901}, {608, 4555}, {664, 8752}, {1022, 7012}, {1119, 5548}, {1398, 4582}, {1417, 6335}, {1877, 4638}, {1880, 4622}, {6548, 7115}
X(14418) = barycentric product X(i)*X(j) for these {i,j}: {3, 4768}, {44, 6332}, {63, 1639}, {69, 4895}, {77, 4528}, {78, 900}, {219, 3762}, {332, 4730}, {345, 1635}, {519, 521}, {522, 5440}, {650, 3977}, {652, 4358}, {905, 2325}, {1332, 4530}, {1459, 4723}, {1647, 4571}, {1812, 4120}, {1946, 3264}, {1960, 3718}, {3689, 4025}
X(14418) = barycentric quotient X(i)/X(j) for these {i,j}: {44, 653}, {78, 4555}, {212, 901}, {219, 3257}, {283, 4622}, {332, 4634}, {521, 903}, {650, 6336}, {652, 88}, {900, 273}, {902, 108}, {1635, 278}, {1639, 92}, {1802, 5548}, {1812, 4615}, {1946, 106}, {1960, 34}, {2193, 4591}, {2325, 6335}, {3063, 8752}, {3251, 1877}, {3689, 1897}, {3692, 4582}, {3762, 331}, {3911, 13149}, {3977, 4554}, {4528, 318}, {4587, 5376}, {4730, 225}, {4768, 264}, {4895, 4}, {5440, 664}, {7004, 6548}, {7117, 1022}, {8611, 4080}

X(14419) =  TRIPOLAR CENTROID OF X(81)

Barycentrics    a*(b-c)*(2*a^2-b^2-c^2) : :

X(14419) lies on the Monge line of the McCay circles (with X(2789)). (Randy Hutson, November 2, 2017)

X(14419) lies on these lines: {1,4730}, {2,2787}, {3,2775}, {10,4922}, {36,238}, {214,3126}, {244,665}, {274,14296}, {351,690}, {650,4378}, {659,764}, {1022,1929}, {1125,4010}, {1960,2254}, {3669,6050}, {3777,4401}, {4160,4367}, {6004,8643}, {6089,11125}, {6366,11124}

X(14419) = isogonal conjugate of X(5380)
X(14419) = complement of X(30709)
X(14419) = X(2642)-cross conjugate of X(4750)
X(14419) = cevapoint of X(351) and X(2642)
X(14419) = crossdifference of every pair of points on line {37, 100}
X(14419) = crosssum of X(i) and X(j) for these (i,j): {513, 7292}, {650, 8540}
X(14419) = crosspoint wrt medial triangle of X(1015) and X(2482)
X(14419) = centroid of (degenerate) side-triangle of ABC and Gemini triangle 23
X(14419) = centroid of (degenerate) cross-triangle of anticevian triangles of X(1) and X(6) (excentral and tangential triangles)
X(14419) = barycentric product X(i)*X(j) for these {i,j}: {1, 4750}, {81, 690}, {86, 2642}, {187, 693}, {274, 351}, {468, 905}, {513, 524}, {514, 896}, {649, 14210}, {650, 7181}, {661, 6629}, {667, 3266}, {876, 4760}, {922, 3261}, {1019, 4062}, {1444, 14273}, {3125, 5468}, {3669, 3712}, {6390, 6591}
X(14419) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 5380}, {81, 892}, {187, 100}, {351, 37}, {468, 6335}, {513, 671}, {524, 668}, {649, 897}, {667, 111}, {690, 321}, {896, 190}, {922, 101}, {1333, 691}, {1648, 4036}, {1919, 923}, {2642, 10}, {3063, 5547}, {3121, 9178}, {3125, 5466}, {3266, 6386}, {3292, 1332}, {3712, 646}, {4062, 4033}, {4750, 75}, {4760, 874}, {5467, 4567}, {5468, 4601}, {6629, 799}, {7181, 4554}, {14210, 1978}
X(14419) = X(i)-isoconjugate of X(j) for these (i,j): {1, 5380}, {10, 691}, {42, 892}, {100, 897}, {101, 671}, {111, 190}, {664, 5547}, {668, 923}, {895, 1897}, {3699, 7316}, {4561, 8753}, {4570, 5466}, {4600, 9178}

X(14420) =  TRIPOLAR CENTROID OF X(83)

Barycentrics    (b-c)*(b+c)*(-2*a^4+b^4+c^4) : :
X(14420) = 3 X(1649) - 2 X(3268) = 4 X(1637) - 3 X(8371) = 3 X(8371) - 2 X(9148) = X(3268) - 3 X(9185) = 5 X(9147) - 3 X(9485) = 3 X(1649) - 4 X(11176) = 3 X(9185) - 2 X(11176)

X(14420) lies on these lines: {2,9479}, {23,385}, {115,125}, {351,2799}, {526,12824}, {804,8029}, {1649,3268}, {4448,6370}, {5466,11606}, {12188,13519}

X(14420) = reflection of X(i) in X(j) for these {i,j}: {1649, 9185}, {3268, 11176}, {8029, 9979}, {9148, 1637}, {11123, 351}
X(14420) = crossdifference of every pair of points on line {39, 110}
X(14420) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1637, 9148, 8371), (3268, 9185, 11176), (3268, 11176, 1649)
X(14420) = X(i)-isoconjugate of X(j) for these (i,j): {58, 5389}, {662, 755}
X(14420) = centroid of (degenerate) cross-triangle of anticevian triangles of X(2) and X(6) (anticomplementary and tangential triangles)
X(14420) = barycentric product X(i)*X(j) for these {i,j}: {514, 4156}, {523, 754}, {850, 8627}, {1577, 2244}, {3700, 7214}, {4157, 7178}
X(14420) = barycentric quotient X(i)/X(j) for these {i,j}: {37, 5389}, {512, 755}, {754, 99}, {2244, 662}, {4156, 190}, {4157, 645}, {7214, 4573}, {8627, 110}

X(14421) =  TRIPOLAR CENTROID OF X(88)

Barycentrics    a*(b-c)*(2*a^2-2*a*b-b^2-2*a*c+4*b*c-c^2) : :
X(14421) = 2 X(1) + X(764) = 2 X(1022) + X(3251) = X(2530) + 2 X(4449) = 4 X(3960) - X(4730) = 4 X(1) - X(6161) = 2 X(764) + X(6161) = 4 X(1022) + X(6161) = X(6161) - 8 X(9269) = X(3251) - 4 X(9269) = X(1022) + 2 X(9269) = X(764) + 4 X(9269)

X(14421) lies on these lines: {1,513}, {100,4618}, {244,665}, {514,551}, {667,999}, {2099,3669}, {2530,4449}, {2832,11716}, {3126,4825}, {3309,10247}, {3679,9260}, {3960,4730}, {4083,5902}, {5376,6163}

X(14421) = midpoint of X(i) and X(j) for these {i,j}: {1, 1022}, {764, 3251}
X(14421) = reflection of X(i) in X(j) for these {i,j}: {1, 9269}, {764, 1022}, {3251, 1}, {4448, 551}, {6161, 3251}
X(14421) = reflection of X(3251) in the line X(1)X(3)
X(14421) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 764, 6161), (1022, 9269, 3251)
X(14421) = X(21)-beth conjugate of X(3251)
X(14421) = X(190)-isoconjugate of X(2384)
X(14421) = X(244)-Hirst inverse of X(1635)
X(14421) = X(4618)-line conjugate of X(100)
X(14421) = crossdifference of every pair of points on line {44, 100}
X(14421) = barycentric product X(i)*X(j) for these {i,j}: {244, 6633}, {513, 545}, {693, 8649}, {1022, 1644}
X(14421) = barycentric quotient X(i)/X(j) for these {i,j}: {545, 668}, {667, 2384}, {6633, 7035}, {8649, 100}

X(14422) =  TRIPOLAR CENTROID OF X(89)

Barycentrics    a*(b-c)*(4*a^2-a*b-2*b^2-a*c+2*b*c-2*c^2) : :
X(14422) = X(1960) + 2 X(3960)

X(14422) lies on these lines: {244,665}, {513,1960}, {551,900}, {659,1022}, {2254,3251}, {2787,4928}, {4378,4893}, {9269,9508}

X(14422) = midpoint of X(i) and X(j) for these {i,j}: {659, 1022}, {2254, 3251}, {4378, 4893}, {9269, 9508}
X(14422) = crossdifference of every pair of points on line {45, 100}
X(14422) = barycentric product X(513)X(4715)
X(14422) = barycentric quotient X(4715)/X(668)

X(14423) =  TRIPOLAR CENTROID OF X(115)

Barycentrics    (b-c)^3*(b+c)^3*(-2*a^2+b^2+c^2)*(-2*a^4+2*a^2*b^2+b^4+2*a^2*c^2-4*b^2*c^2+c^4) : :
X(14423) = X(1641) - 3 X(8371)

X(14423) lies on these lines: {524,10278}, {671,690}, {1641,8371}, {1648,8029}, {10189,11053}

X(14423) = midpoint of X(1648) and X(8029)
X(14423) = reflection of X(11053) in X(10189)
X(14423) = crossdifference of every pair of points on line {249, 843}
X(14423) = barycentric product X(i)*X(j) for these {i,j}: {1641, 8029}, {1648, 8371}
X(14423) = barycentric quotient X(1648)/X(9170)

X(14424) =  TRIPOLAR CENTROID OF X(141)

Barycentrics    (b-c) (b+c) (b^2+c^2) (-2 a^2+b^2+c^2) : :
X(14424) = 2 X(351) - 3 X(1649) = 4 X(2525) - X(2528) = 2 X(2525) + X(3005) = X(2528) + 2 X(3005) = X(669) - 4 X(3265) = 5 X(3005) - 2 X(3806) = 5 X(2525) + X(3806) = 5 X(2528) + 4 X(3806) = 5 X(351) - 6 X(9125) = 5 X(1649) - 4 X(9125) = 3 X(3268) - X(9131) = 3 X(8029) - 4 X(9134) = 2 X(9134) - 3 X(9148) = 3 X(8371) - 2 X(9979) = 3 X(9191) - X(9979) = 2 X(9131) - 3 X(11123)

X(14424) lies on these lines: {2,9479}, {351,690}, {523,7840}, {669,3265}, {804,3268}, {826,2474}, {1634,4576}, {2799,8029}, {3800,8665}, {8371,9191}, {9147,10190}, {9178,14279}

X(14424) = reflection of X(i) in X(j) for these {i,j}: {8029, 9148}, {8371, 9191}, {9147, 10190}, {9178, 14279}, {11123, 3268}
X(14424) = anticomplement of X(32193)
X(14424) = X(3266)-Ceva conjugate of X(1648)
X(14424) = crosspoint of X(67) and X(99)
X(14424) = crossdifference of every pair of points on line {111, 251}
X(14424) = crosssum of X(23) and X(512)
X(14424) = {X(2525),X(3005)}-harmonic conjugate of X(2528)
X(14424) = X(i)-isoconjugate of X(j) for these (i,j): {82, 691}, {111, 4599}, {827, 897}, {923, 4577}
X(14424) = barycentric product X(i)*X(j) for these {i,j}: {141, 690}, {351, 8024}, {468, 2525}, {523, 7813}, {524, 826}, {1648, 4576}, {1930, 2642}, {3005, 3266}, {3933, 14273}, {8061, 14210}
X(14424) = barycentric quotient X(i)/X(j) for these {i,j}: {39, 691}, {141, 892}, {187, 827}, {351, 251}, {524, 4577}, {690, 83}, {826, 671}, {896, 4599}, {2084, 923}, {2642, 82}, {3005, 111}, {3266, 689}, {3954, 5380}, {7813, 99}, {8061, 897}, {14210, 4593}

X(14425) =  TRIPOLAR CENTROID OF X(145)

Barycentrics    (2 a-b-c) (3 a-b-c) (b-c) : :
X(14425) = X(650) + 2 X(2490) = X(661) + 2 X(2527) = X(676) + 2 X(2977) = 2 X(2516) + X(3239) = 3 X(1639) - X(4120) = 3 X(1635) + X(4120) = 2 X(2487) + X(4468) = X(4394) + 2 X(4521) = 2 X(1960) + X(4528) = 3 X(1635) - X(4773) = 3 X(1639) + X(4773) = X(2976) + 2 X(4925) = 10 X(4521) - X(4949) = 5 X(4394) + X(4949) = 7 X(4120) - 3 X(4958) = 7 X(1639) - X(4958) = 7 X(1635) + X(4958) = 7 X(4773) + 3 X(4958) = 5 X(4773) - 3 X(4984) = 5 X(1635) - X(4984) = 5 X(1639) + X(4984) = 5 X(4120) + 3 X(4984) = 5 X(4958) + 7 X(4984) = X(4120) - 9 X(6544) = X(1639) - 3 X(6544) = X(1635) + 3 X(6544) = X(4773) + 9 X(6544) = X(4984) + 15 X(6544) = 10 X(2490) - X(6590) = 5 X(650) + X(6590) = 2 X(4949) - 5 X(14321) = 4 X(4521) - X(14321) = 2 X(4394) + X(14321) = 5 X(14321) - 8 X(14350) = X(4949) - 4 X(14350) = 5 X(4521) - 2 X(14350) = 5 X(4394) + 4 X(14350) = 7 X(4394) - 4 X(14351) = 7 X(4521) + 2 X(14351) = 7 X(14350) + 5 X(14351) = 7 X(14321) + 8 X(14351)

X(14425) lies on these lines: {2,4927}, {230,231}, {375,6373}, {649,4943}, {661,2527}, {900,1635}, {918,4763}, {1638,6546}, {1960,4528}, {2487,4468}, {2516,3239}, {2826,3035}, {2976,3667}, {3756,4534}, {4893,4977}, {4928,6009}, {4931,4976}

X(14425) = midpoint of X(i) and X(j) for these {i,j}: {1635, 1639}, {1638, 6546}, {4120, 4773}, {4763, 10196}, {4931, 4976}
X(14425) = complement X(4927)
X(14425) = X(i)-complementary conjugate of X(j) for these (i,j): {31, 5516}, {6079, 2887}
X(14425) = X(i)-Ceva conjugate of X(j) for these (i,j): {2, 5516}, {2403, 3667}, {2415, 519}
X(14425) = crosspoint of X(i) and X(j) for these (i,j): {2, 6079}, {519, 2415}, {2403, 3667}
X(14425) = crossdifference of every pair of points on line {3, 106}
X(14425) = crosssum of X(i) and X(j) for these (i,j): {6, 6085}, {106, 2441}, {1293, 2429}
X(14425) = X(i)-isoconjugate of X(j) for these (i,j): {88, 1293}, {679, 2429}, {901, 8056}, {3257, 3445}
X(14425) = barycentric product X(i)*X(j) for these {i,j}: {44, 4462}, {145, 900}, {513, 4487}, {519, 3667}, {1420, 4768}, {1639, 5435}, {1743, 3762}, {2403, 4370}, {3264, 8643}, {3911, 4521}, {4358, 4394}, {5516, 6079}
X(14425) = barycentric quotient X(i)/X(j) for these {i,j}: {145, 4555}, {900, 4373}, {902, 1293}, {1017, 2429}, {1023, 5382}, {1635, 8056}, {1639, 6557}, {1743, 3257}, {1960, 3445}, {2441, 2226}, {3052, 901}, {3161, 4582}, {3667, 903}, {3756, 6548}, {4120, 4052}, {4162, 1320}, {4370, 2415}, {4394, 88}, {4487, 668}, {4521, 4997}, {4528, 6556}, {4729, 4674}, {4895, 3680}, {5516, 4927}, {8643, 106}, {14321, 4080}

X(14426) =  TRIPOLAR CENTROID OF X(192)

Barycentrics    a (b-c) (a b+a c-2 b c) (a b+a c-b c) : :
X(14426) = X(4507) + 2 X(4940)

X(14426) lies on these lines: {2,6373}, {512,4776}, {513,4763}, {661,665}, {890,899}, {891,4728}, {3123,6377}, {3808,10196}, {3835,4083}, {4010,4768}, {4014,9283}, {4507,4940}

X(14426) = X(715)-complementary conjugate of X(244)
X(14426) = X(3768)-Ceva conjugate of X(891)
X(14426) = crossdifference of every pair of points on line {739, 932}
X(14426) = centroid of (degenerate) side-triangle of ABC and Gemini triangle 16
X(14426) = X(i)-isoconjugate of X(j) for these (i,j): {87, 898}, {739, 4598}, {889, 7121}, {2162, 4607}
X(14426) = barycentric product X(i)*X(j) for these {i,j}: {43, 4728}, {192, 891}, {536, 4083}, {890, 6382}, {899, 3835}, {3212, 4526}, {3768, 6376}
X(14426) = barycentric quotient X(i)/X(j) for these {i,j}: {43, 4607}, {192, 889}, {890, 2162}, {891, 330}, {899, 4598}, {2176, 898}, {3230, 932}, {3768, 87}, {4083, 3227}, {4526, 7155}, {4728, 6384}, {8640, 739}

X(14427) =  TRIPOLAR CENTROID OF X(200)

Barycentrics    a (a-b-c)^2 (2 a-b-c) (b-c) : :
X(14427) = X(657) + 2 X(4130) = 4 X(4130) - X(4171) = 2 X(657) + X(4171) = 5 X(657) - 2 X(4827) = 5 X(4130) + X(4827) = 5 X(4171) + 4 X(4827)

X(14427) lies on these lines: {9,3738}, {45,1769}, {346,4148}, {657,3900}, {900,1635}, {2310,3119}, {2325,4768}

X(14427) = {X(657),X(4130)}-harmonic conjugate of X(4171)
X(14427) = X(i)-Ceva conjugate of X(j) for these (i,j): {1023, 3689}, {1639, 4895}
X(14427) = crosspoint of X(i) and X(j) for these (i,j): {1023, 3689}, {1639, 4528}
X(14427) = X(14427) = crossdifference of every pair of points on line {106, 269}
X(14427) = X(i)-isoconjugate of X(j) for these (i,j): {88, 934}, {106, 658}, {269, 3257}, {279, 901}, {479, 5548}, {903, 1461}, {1022, 7045}, {1042, 4615}, {1262, 6548}, {1320, 4617}, {1407, 4555}, {1417, 4554}, {1427, 4622}, {2316, 4626}, {3668, 4591}, {4569, 9456}, {4582, 7023}, {4619, 6549}, {4637, 4674}, {4997, 6614}
X(14427) = barycentric product X(i)*X(j) for these {i,j}: {1, 4528}, {8, 4895}, {9, 1639}, {44, 3239}, {55, 4768}, {200, 900}, {220, 3762}, {341, 1960}, {346, 1635}, {519, 3900}, {522, 3689}, {644, 4530}, {650, 2325}, {657, 4358}, {663, 4723}, {902, 4397}, {1021, 3943}, {1023, 1146}, {1043, 4730}, {1319, 4163}, {1320, 4543}, {1647, 4578}, {2087, 6558}, {2287, 4120}, {3264, 8641}, {3911, 4130}
X(14427) = barycentric quotient X(i)/X(j) for these {i,j}: {44, 658}, {200, 4555}, {220, 3257}, {519, 4569}, {657, 88}, {728, 4582}, {900, 1088}, {902, 934}, {1023, 1275}, {1043, 4634}, {1253, 901}, {1319, 4626}, {1404, 4617}, {1635, 279}, {1639, 85}, {1960, 269}, {2251, 1461}, {2287, 4615}, {2310, 6548}, {2325, 4554}, {2328, 4622}, {3285, 4637}, {3689, 664}, {3900, 903}, {4105, 1320}, {4120, 1446}, {4130, 4997}, {4171, 4080}, {4524, 4674}, {4528, 75}, {4723, 4572}, {4730, 3668}, {4768, 6063}, {4895, 7}, {6602, 5548}, {8641, 106}, {8756, 13149}

X(14428) =  TRIPOLAR CENTROID OF X(251)

Barycentrics    a^2 (b-c) (b+c) (2 a^4-b^4-c^4) : :
X(14428) = 2 X(2492) + X(5027) = X(6) + 2 X(5113) = 2 X(351) + X(9171) = 2 X(9188) + X(9208) = 2 X(2485) + X(9426) = X(182) + 2 X(11620)

X(14428) lies on these lines: {6,5113}, {182,11620}, {351,865}, {512,1691}, {526,6593}, {2492,2872}, {9035,11176}

X(14428) = crossdifference of every pair of points on line {99, 141}
X(14428) = X(i)-isoconjugate of X(j) for these (i,j): {86, 5389}, {755, 799}
X(14428) = barycentric product X(i)*X(j) for these {i,j}: {512, 754}, {523, 8627}, {649, 4156}, {661, 2244}, {3709, 7214}, {4157, 7180}
X(14428) = barycentric quotient X(i)/X(j) for these {i,j}: {213, 5389}, {669, 755}, {754, 670}, {2244, 799}, {4156, 1978}, {8627, 99}

X(14429) =  TRIPOLAR CENTROID OF X(306)

Barycentrics    (b-c) (b+c) (-2 a+b+c) (-a^2+b^2+c^2) : :
X(14429) = 2 X(656) + X(4064)

X(14429) lies on these lines: {2,11125}, {4,9524}, {121,4768}, {122,125}, {525,656}, {900,1635}, {1459,9031}, {2773,5692}

X(14429) = reflection of X(11125) in X(2)
X(14429) = crossdifference of every pair of points on line {106, 112}
X(14429) = X(i)-isoconjugate of X(j) for these (i,j): {19, 4591}, {25, 4622}, {28, 901}, {88, 112}, {106, 162}, {163, 6336}, {648, 9456}, {662, 8752}, {1396, 5548}, {1474, 3257}, {1973, 4615}, {1974, 4634}, {2203, 4555}, {4246, 10428}
X(14429) = barycentric product X(i)*X(j) for these {i,j}: {44, 14208}, {69, 4120}, {72, 3762}, {304, 4730}, {306, 900}, {307, 1639}, {519, 525}, {523, 3977}, {647, 3264}, {656, 4358}, {902, 3267}, {905, 3992}, {1214, 4768}, {1231, 4895}, {1565, 4169}, {1577, 5440}, {3265, 8756}, {3943, 4025}
X(14429) = barycentric quotient X(i)/X(j) for these {i,j}: {3, 4591}, {44, 162}, {63, 4622}, {69, 4615}, {71, 901}, {72, 3257}, {125, 4049}, {304, 4634}, {306, 4555}, {512, 8752}, {519, 648}, {520, 1797}, {523, 6336}, {525, 903}, {647, 106}, {656, 88}, {810, 9456}, {900, 27}, {902, 112}, {1023, 5379}, {1635, 28}, {1639, 29}, {1960, 1474}, {2318, 5548}, {3264, 6331}, {3710, 4582}, {3762, 286}, {3943, 1897}, {3977, 99}, {3992, 6335}, {4064, 4080}, {4120, 4}, {4358, 811}, {4466, 6548}, {4528, 2322}, {4574, 9268}, {4730, 19}, {4895, 1172}, {5440, 662}, {8611, 1320}, {8756, 107}

X(14430) =  TRIPOLAR CENTROID OF X(312)

Barycentrics    (b-c) (-a+b+c) (-a b-a c+2 b c) : :
X(14430) = X(10) - X(2254) = X(8) + 2 X(3716) = 2 X(10) + X(3762) = X(2254) + 2 X(3762) = 5 X(1698) - 2 X(3960) = X(4041) - 4 X(4147) = 2 X(4147) + X(4391) = X(4041) + 2 X(4391) = 2 X(650) + X(4474) = 4 X(4791) - X(4804) = 4 X(3716) - X(4895) = 2 X(8) + X(4895) = 2 X(4397) + X(6615) = X(4088) + 2 X(10015)

X(14430) lies on these lines: {8,3716}, {10,2254}, {11,1146}, {522,3717}, {650,4474}, {891,4728}, {958,8648}, {1635,2787}, {1698,3960}, {2533,4977}, {2789,10196}, {2814,5587}, {2832,10712}, {3679,3887}, {4088,10015}, {4490,4802}, {4791,4804}

X(14430) = X(4526)-cross conjugate of X(4728)
X(14430) = crossdifference of every pair of points on line {109, 604}
X(14430) = X(1639)-Hirst inverse of X(4124)
X(14430) = centroid of (degenerate) side-triangle of ABC and Gemini triangle 13
X(14430) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (8, 3716, 4895), (10, 3762, 2254), (4147, 4391, 4041)
X(14430) = X(i)-isoconjugate of X(j) for these (i,j): {56, 898}, {604, 4607}, {651, 739}, {889, 1397}, {1415, 3227}
X(14430) = barycentric product X(i)*X(j) for these {i,j}: {8, 4728}, {75, 4526}, {312, 891}, {514, 4009}, {522, 536}, {650, 6381}, {899, 4391}, {3596, 3768}, {3994, 4560}
X(14430) = barycentric quotient X(i)/X(j) for these {i,j}: {8, 4607}, {9, 898}, {312, 889}, {522, 3227}, {536, 664}, {663, 739}, {890, 604}, {891, 57}, {899, 651}, {3230, 109}, {3699, 5381}, {3768, 56}, {3994, 4552}, {4009, 190}, {4526, 1}, {4728, 7}, {6381, 4554}

X(14431) =  TRIPOLAR CENTROID OF X(321)

Barycentrics    (b-c) (b+c) (-a b-a c+2 b c) : :
X(14431) = X(764) + 2 X(3762) = X(764) - 4 X(3837) = X(3762) + 2 X(3837) = 2 X(10) + X(4010) = X(2533) + 2 X(4129) = X(2530) + 2 X(4391) = 2 X(1577) + X(4705) = 4 X(10) - X(4730) = 2 X(4010) + X(4730) = X(1491) + 2 X(4791) = 2 X(4770) + X(4804) = X(4761) + 2 X(4806) = X(4490) + 2 X(4823) = X(4378) - 4 X(4885) = 4 X(1125) - X(4922) = 4 X(4129) - X(4983) = 2 X(2533) + X(4983) = 4 X(3716) - X(6161) = 5 X(1698) - 2 X(9508) = 4 X(6702) - X(13277)

X(14431) lies on these lines: {2,2787}, {10,4010}, {115,125}, {119,120}, {381,2775}, {523,1577}, {764,3762}, {891,4728}, {1125,4922}, {1491,4791}, {1698,9508}, {1734,4926}, {2530,4391}, {2533,4129}, {2789,11814}, {3716,6161}, {3887,4800}, {4013,4049}, {4378,4885}, {4490,4823}, {4761,4806}, {4770,4804}, {5466,11611}, {6702,13277}

X(14431) = crossdifference of every pair of points on line {110, 739}
X(14431) = centroid of (degenerate) side-triangle of ABC and Gemini triangle 21
X(14431) = centroid of (degenerate) side-triangle of ABC and Gemini triangle 27
X(14431) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (10, 4010, 4730), (2533, 4129, 4983), (3762, 3837, 764)
X(14431) = X(i)-isoconjugate of X(j) for these (i,j): {58, 898}, {163, 3227}, {662, 739}, {889, 2206}, {1333, 4607}
X(14431) = barycentric product X(i)*X(j) for these {i,j}: {10, 4728}, {313, 3768}, {321, 891}, {514, 3994}, {523, 536}, {661, 6381}, {850, 3230}, {899, 1577}, {1441, 4526}, {4009, 7178}
X(14431) = barycentric quotient X(i)/X(j) for these {i,j}: {10, 4607}, {37, 898}, {321, 889}, {512, 739}, {523, 3227}, {536, 99}, {890, 1333}, {891, 81}, {899, 662}, {1646, 3733}, {3230, 110}, {3768, 58}, {3952, 5381}, {3994, 190}, {4009, 645}, {4526, 21}, {4728, 86}, {6381, 799}

X(14432) =  TRIPOLAR CENTROID OF X(333)

Barycentrics    (a-b-c) (b-c) (2 a^2-b^2-c^2) : :
X(14432) = 2 X(3716) + X(3904) = 2 X(2605) + X(4064) = 2 X(1) + X(4088) = 5 X(3616) - 2 X(4458) = 4 X(3239) - X(4474) = 4 X(1125) - X(4707) = X(663) + 2 X(6332)

X(14432) lies on these lines: {1,4088}, {2,2785}, {11,1146}, {351,690}, {522,663}, {891,6546}, {1125,4707}, {2605,4064}, {2774,5692}, {2787,4120}, {3239,4474}, {3616,4458}, {3716,3904}, {4984,8632}, {6370,11125}, {7628,8058}

X(14432) = crossdifference of every pair of points on line {109, 111}
X(14432) = crosspoint wrt medial triangle of X(1146) and X(2482)
X(14432) = centroid of (degenerate) side-triangle of ABC and Gemini triangle 1
X(14432) = X(i)-isoconjugate of X(j) for these (i,j): {56, 5380}, {65, 691}, {100, 7316}, {108, 895}, {109, 897}, {111, 651}, {664, 923}, {671, 1415}, {892, 1402}, {934, 5547}, {6516, 8753}
X(14432) = barycentric product X(i)*X(j) for these {i,j}: {8, 4750}, {314, 2642}, {332, 14273}, {333, 690}, {468, 6332}, {514, 3712}, {522, 524}, {650, 14210}, {663, 3266}, {896, 4391}, {3064, 6390}, {3239, 7181}, {3700, 6629}, {4062, 4560}
X(14432) = barycentric quotient X(i)/X(j) for these {i,j}: {9, 5380}, {187, 109}, {284, 691}, {333, 892}, {351, 1400}, {468, 653}, {522, 671}, {524, 664}, {649, 7316}, {650, 897}, {652, 895}, {657, 5547}, {663, 111}, {690, 226}, {896, 651}, {922, 1415}, {2642, 65}, {3063, 923}, {3266, 4572}, {3292, 1813}, {3712, 190}, {4062, 4552}, {4750, 7}, {5468, 4620}, {6629, 4573}, {7181, 658}, {7267, 6649}, {14210, 4554}, {14273, 225}

X(14433) =  TRIPOLAR CENTROID OF X(350)

Barycentrics    (b-c) (-a^2+b c) (-a b-a c+2 b c) : :

X(14433) lies on these lines: {2,514}, {210,4083}, {239,4375}, {513,4688}, {649,4384}, {659,4508}, {812,3766}, {891,4728}, {3661,3835}, {4124,4448}, {6372,8027}

X(14433) = crossdifference of every pair of points on line {739, 813}
X(14433) = X(891)-Hirst inverse of X(4728)
X(14433) = X(i)-isoconjugate of X(j) for these (i,j): {292, 898}, {660, 739}, {875, 5381}, {889, 1922}, {1911, 4607}
X(14433) = barycentric product X(i)*X(j) for these {i,j}: {239, 4728}, {350, 891}, {514, 4465}, {536, 812}, {659, 6381}, {899, 3766}, {1921, 3768}, {4526, 10030}
X(14433) = barycentric quotient X(i)/X(j) for these {i,j}: {238, 898}, {239, 4607}, {350, 889}, {536, 4562}, {812, 3227}, {890, 1911}, {891, 291}, {899, 660}, {1646, 3572}, {3230, 813}, {3570, 5381}, {3768, 292}, {4465, 190}, {4526, 4876}, {4728, 335}, {6381, 4583}, {8632, 739}

X(14434) =  TRIPOLAR CENTROID OF X(536)

Barycentrics    a (b-c) (a b+a c-2 b c)^2 : :
X(14434) = 4 X(2) - X(8027)

X(14434) lies on the cubic K219 and these lines: {2,513}, {10,3835}, {661,764}, {693,6376}, {891,4728}

X(14434) = X(i)-complementary conjugate of X(j) for these (i,j): {31, 1646}, {100, 4871}, {101, 536}, {536, 116}, {765, 891}, {899, 11}, {1252, 4763}, {3230, 1086}, {3768, 6547}, {3994, 125}, {4009, 124}
X(14434) = X(i)-Ceva conjugate of X(j) for these (i,j): {2, 1646}, {513, 891}, {668, 536}
X(14434) = crosspoint of X(i) and X(j) for these (i,j): {513, 891}, {536, 668}
X(14434) = crossdifference of every pair of points on line {739, 898}
X(14434) = crosssum of X(i) and X(j) for these (i,j): {100, 898}, {667, 739}
X(14434) = X(739)-isoconjugate of X(4607)
X(14434) = barycentric product X(i)*X(j) for these {i,j}: {513, 13466}, {536, 891}, {899, 4728}, {3768, 6381}
X(14434) = barycentric quotient X(i)/X(j) for these {i,j}: {536, 889}, {890, 739}, {891, 3227}, {899, 4607}, {3230, 898}, {13466, 668}

X(14435) =  TRIPOLAR CENTROID OF X(551)

Barycentrics    (b-c) (-2 a+b+c) (4 a+b+c) : :
X(14435) = 5 X(1635) - 2 X(1639) = 8 X(1639) - 5 X(4120) = 4 X(1635) - X(4120) = 5 X(649) + 4 X(4765) = X(1635) + 2 X(4773) = X(1639) + 5 X(4773) = X(4120) + 8 X(4773) = 10 X(4394) - X(4820) = 14 X(1639) - 5 X(4958) = 7 X(4120) - 4 X(4958) = 7 X(1635) - X(4958) = 14 X(4773) + X(4958) = 4 X(4773) - X(4984) = 2 X(1635) + X(4984) = X(4120) + 2 X(4984) = 4 X(1639) + 5 X(4984) = 2 X(4958) + 7 X(4984) = 8 X(649) + X(4988) = 2 X(4958) - 7 X(6544) = 4 X(1639) - 5 X(6544) = 4 X(4773) + X(6544) = 7 X(649) - 16 X(14351)

X(14435) lies on these lines: {239,514}, {900,1635}, {4394,4820}, {4730,8661}, {4893,6006}, {6009,6545}

X(14435) = midpoint of X(4984) and X(6544)
X(14435) = reflection of X(i) in X(j) for these {i,j}: {4120, 6544}, {6544, 1635}
X(14435) = crossdifference of every pair of points on line {42, 106}
X(14435) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1635, 4773, 4984), (1635, 4984, 4120)
X(14435) = barycentric product X(i)*X(j) for these {i,j}: {551, 900}, {1639, 4031}, {1647, 4781}
X(14435) = barycentric quotient X(i)/X(j) for these {i,j}: {551, 4555}, {3707, 4582}

X(14436) =  TRIPOLAR CENTROID OF X(869)

Barycentrics    a^3 (2 a-b-c) (b-c) (b^2+b c+c^2) : :

X(14436) lies on these lines: {798,890}, {900,1635}, {1919,8645}, {2484,5075}

X(14436) = crossdifference of every pair of points on line {106, 789}
X(14436) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1635, 4773, 4984), (1635, 4984, 4120)
X(i)-isoconjugate of X(j) for these (i,j): {88, 789}, {870, 3257}, {903, 4586}
X(14436) = barycentric product X(i)*X(j) for these {i,j}: {44, 3250}, {519, 788}, {667, 4439}, {824, 2251}, {869, 900}, {902, 1491}, {984, 1960}, {1469, 4895}, {1635, 2276}, {3264, 8630}, {3736, 4730}
X(14436) = barycentric quotient X(i)/X(j) for these {i,j}: {788, 903}, {869, 4555}, {900, 871}, {902, 789}, {1023, 5388}, {1960, 870}, {2251, 4586}, {3736, 4634}, {4439, 6386}, {8630, 106}, {9459, 1492}

X(14437) =  TRIPOLAR CENTROID OF X(899)

Barycentrics    a (2 a-b-c) (b-c) (a b+a c-2 b c) : :

X(14437) lies on these lines: {37,513}, {45,649}, {190,4375}, {514,4664}, {891,3768}, {900,1635}, {1919,2267}, {2087,3251}, {2276,4893}, {3835,4389}

X(14437) = crossdifference of every pair of points on line {106, 238}
X(14437) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1635, 4773, 4984), (1635, 4984, 4120)
X(14437) = X(i)-isoconjugate of X(j) for these (i,j): {88, 898}, {106, 4607}, {739, 4555}, {889, 9456}, {901, 3227}
X(14437) = barycentric product X(i)*X(j) for these {i,j}: {44, 4728}, {519, 891}, {536, 1635}, {890, 3264}, {899, 900}, {1960, 6381}, {3230, 3762}, {3768, 4358}, {3911, 4526}
X(14437) = barycentric quotient X(i)/X(j) for these {i,j}: {44, 4607}, {519, 889}, {890, 106}, {891, 903}, {899, 4555}, {902, 898}, {1023, 5381}, {1635, 3227}, {1646, 1022}, {3230, 3257}, {3768, 88}, {4526, 4997}

X(14438) =  TRIPOLAR CENTROID OF X(985)

Barycentrics    a (b-c) (2 a^3-b^3-c^3) : :

X(14438) lies on these lines: {244,665}, {513,1919}, {649,3960}, {661,1960}, {4160,4893}, {4979,8658}

X(14438) = isogonal conjugate of X(5386)
X(14438) = crossdifference of every pair of points on line {100, 753}
X(14438) = X(i)-isoconjugate of X(j) for these (i,j): {1, 5386}, {190, 753}
X(14438) = barycentric product X(i)*X(j) for these {i,j}: {1, 4809}, {513, 752}, {514, 2243}, {693, 8626}, {1019, 4144}, {3669, 4070}
X(14438) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 5386}, {667, 753}, {752, 668}, {2243, 190}, {4070, 646}, {4144, 4033}, {4809, 75}, {8626, 100}

X(14439) =  TRIPOLAR CENTROID OF X(1026)

Barycentrics    a (2 a-b-c) (a b-b^2+a c-c^2) : :
X(14439) = 2 X(1018) + X(2170) = X(672) + 2 X(3693) = 4 X(3693) - X(3930) = 2 X(672) + X(3930) = X(4712) + 2 X(8299)

X(14439) lies on these lines: {1,4752}, {6,3722}, {9,100}, {37,244}, {44,678}, {45,4413}, {190,9318}, {214,1023}, {392,1334}, {518,672}, {665,1642}, {900,1635}, {1018,2170}, {1054,3731}, {1145,4530}, {1475,3991}, {2325,3911}, {3247,3315}, {3730,5692}, {4141,4908}, {4169,4738}, {5030,5525}

X(14439) = X(i)-beth conjugate of X(j) for these (i,j): {9, 244}, {644, 2802}
X(14439) = X(44)-Hirst inverse of X(3689)
X(14439) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (9, 100, 2246), (672, 3693, 3930)
X(14439) = X(i)-isoconjugate of X(j) for these (i,j): {88, 105}, {106, 673}, {903, 1438}, {919, 6548}, {1027, 3257}, {1320, 1462}, {1416, 4997}, {2481, 9456}
X(14439) = crossdifference of every pair of points on line {105, 106}
X(14439) = barycentric product X(i)*X(j) for these {i,j}: {44, 3912}, {241, 2325}, {518, 519}, {672, 4358}, {883, 4895}, {900, 1026}, {902, 3263}, {918, 1023}, {1025, 1639}, {1319, 3717}, {1458, 4723}, {1861, 5440}, {2223, 3264}, {2283, 4768}, {2284, 3762}, {3286, 3992}, {3689, 9436}, {3693, 3911}, {3977, 5089}
X(14439) = barycentric quotient X(i)/X(j) for these {i,j}: {44, 673}, {518, 903}, {519, 2481}, {665, 1022}, {672, 88}, {902, 105}, {1023, 666}, {1026, 4555}, {1404, 1462}, {1960, 1027}, {2223, 106}, {2251, 1438}, {2254, 6548}, {2284, 3257}, {2340, 1320}, {3675, 6549}, {3693, 4997}, {3930, 4080}, {4895, 885}, {5089, 6336}, {9454, 9456}

X(14440) =  TRIPOLAR CENTROID OF X(1267)

Barycentrics    a (b-c) (a b+a c-2 b c) (b c+S) : :

X(14440) lies on these lines: {891,4728}, {4131,6364}

X(14440) = crossdifference of every pair of points on line {739, 6135}
X(14440) = barycentric product X(i)*X(j) for these {i,j}: {536, 6364}, {891, 1267}, {3083, 4728}, {4526, 13453}
X(14440) = barycentric quotient X(i)/X(j) for these {i,j}: {891, 1123}, {1124, 898}, {1267, 889}, {3083, 4607}, {3230, 6135}, {4526, 13454}, {6364, 3227}

X(14441) =  TRIPOLAR CENTROID OF X(1646)

Barycentrics    a^3 (b-c)^3 (a b+a c-2 b c)^2:; X(668) + 2 X(9267)

X(14441) lies on these lines: {513,3227}, {668,9267}, {764,3123}, {891,4728}, {1015,8027}, {3251,6373}

X(14441) = reflection of X(8027) in X(1015)
X(14441) = X(i)-Ceva conjugate of X(j) for these (i,j): {513, 1646}, {9267, 891}
X(14441) = crosspoint of X(i) and X(j) for these (i,j): {513, 1646}, {891, 1015}
X(14441) = crossdifference of every pair of points on line {739, 5381}
X(14441) = crosssum of X(i) and X(j) for these (i,j): {100, 5381}, {898, 1016}
X(14441) = X(4607)-isoconjugate of X(5381)
X(14441) = barycentric product X(i)*X(j) for these {i,j}: {891, 1646}, {8027, 13466}
X(14441) = barycentric quotient X(i)/X(j) for these {i,j}: {890, 5381}, {1646, 889}

X(14442) =  TRIPOLAR CENTROID OF X(1647)

Barycentrics    (b-c)^3 (-2 a+b+c)^2 : :
X(14442) = 2 X(4370) - 3 X(6544)

X(14442) lies on these lines: {190,6634}, {514,903}, {545,6546}, {649,2161}, {900,1635}, {909,1919}, {1086,6545}, {3667,12034}

X(14442) = reflection of X(i) in X(j) for these {i,j}: {190, 10196}, {6545, 1086}
X(14442) = X(i)-Ceva conjugate of X(j) for these (i,j): {514, 1647}, {6545, 6550}, {6632, 519}
X(14442) = X(i)-isoconjugate of X(j) for these (i,j): {88, 6551}, {765, 4638}, {901, 5376}, {1252, 4618}, {3257, 9268}, {6635, 9456}
X(14442) = crosspoint of X(i) and X(j) for these (i,j): {514, 1647}, {519, 6632}, {900, 1086}, {6545, 6550}
X(14442) = X(14442) = crossdifference of every pair of points on line {106, 6551}
X(14442) = crosssum of X(i) and X(j) for these (i,j): {101, 9268}, {901, 1252}
X(14442) = barycentric product X(i)*X(j) for these {i,j}: {519, 6550}, {522, 14027}, {764, 4738}, {900, 1647}, {1086, 6544}, {1111, 3251}, {1358, 4543}, {2087, 3762}, {3264, 8661}, {3676, 4542}, {4370, 6545}
X(14442) = barycentric quotient X(i)/X(j) for these {i,j}: {244, 4618}, {519, 6635}, {764, 679}, {902, 6551}, {1015, 4638}, {1635, 5376}, {1647, 4555}, {1960, 9268}, {2087, 3257}, {3251, 765}, {4370, 6632}, {4530, 4582}, {4542, 3699}, {4543, 4076}, {6544, 1016}, {6550, 903}, {8661, 106}, {14027, 664}

X(14443) =  TRIPOLAR CENTROID OF X(1648)

Barycentrics    (b-c)^3 (b+c)^3 (-2 a^2+b^2+c^2)^2 : :
X(14443) = 3 X(1649) - 2 X(2482) = 4 X(5461) - 3 X(8371) = X(8591) - 3 X(9168) = X(671) - 3 X(9180) = X(99) + 2 X(9293) = 3 X(9166) - 2 X(10278) = 5 X(8029) - 4 X(12076) = 5 X(115) - 2 X(12076) = 4 X(10189) - 5 X(14061) = 2 X(13187) - 5 X(14061)

X(14443) lies on these lines: {99 =9293}, {115,8029}, {351,690}, {523,671}, {543,11123}, {669,3455}, {1576,14366}, {5461,8371}, {5465,13291}, {5466,9183}, {8591,9168}, {9166,10278}, {10189,13187}

X(14443) = midpoint of X(9293) and X(10190)
X(14443) = reflection of X(i) in X(j) for these {i,j}: {99, 10190}, {5466, 9183}, {8029, 115}, {13187, 10189}, {13291, 5465}
X(14443) = X(i)-Ceva conjugate of X(j) for these (i,j): {523, 1648}, {9293, 690}
X(14443) = crosspoint of X(i) and X(j) for these (i,j): {115, 690}, {523, 1648}
X(14443) = crosssum of X(249) and X(691)
X(14443) = barycentric product X(i)*X(j) for these {i,j}: {115, 1649}, {690, 1648}, {2482, 8029}
X(14443) = barycentric quotient X(i)/X(j) for these {i,j}: {1648, 892}, {1649, 4590}

X(14444) =  TRIPOLAR CENTROID OF X(1649)

Barycentrics    (b-c)^2 (b+c)^2 (-2 a^2+b^2+c^2)^3 : :
X(14444) = 4 X(620) - 3 X(1641) = 2 X(115) - 3 X(1648)

X(14444) lies on these lines: {6,11006}, {99,524}, {115,125}, {620,1641}, {2482,8030}

X(14444) = reflection of X(8030) in X(2482)
X(14444) = X(524)-Ceva conjugate of X(1649)
X(14444) = crosspoint of X(i) and X(j) for these (i,j): {524, 1649}, {690, 2482}
X(14444) = crossdifference of every pair of points on line {110, 9171}
X(14444) = X(14444) = crosssum of X(691) and X(10630)
X(14444) = barycentric product X(i)*X(j) for these {i,j}: {115, 8030}, {690, 1649}, {1648, 2482}
X(14444) = barycentric quotient X(i)/X(j) for these {i,j}: {1649, 892}, {8030, 4590}

X(14445) =  TRIPOLAR CENTROID OF X(5391)

Barycentrics    a (b-c) (a b+a c-2 b c) (b c-S) : :

X(14445) lies on these lines: {891,4728}, {4131,6365}

X(14445) = crossdifference of every pair of points on line {739, 6136}
X(14445) = barycentric product X(i)*X(j) for these {i,j}: {536, 6365}, {891, 5391}, {3084, 4728}, {4526, 13436}
X(14445) = barycentric quotient X(i)/X(j) for these {i,j}: {891, 1336}, {1335, 898}, {3084, 4607}, {3230, 6136}, {4526, 13426}, {5391, 889}, {6365, 3227}

X(14446) =  TRIPOLAR CENTROID OF X(17)

Trilinears    Sin(A-Pi/3) Sin(B-C) (Cos(B-C)+2 Cos(A-Pi/3)) : :
Barycentrics    (b^2-c^2) (2 a^4-a^2 b^2-b^4-a^2 c^2+2 b^2 c^2-c^4+2 Sqrt(3) (2 a^2-b^2-c^2) S) : :
X(14446) = 2 X(1637) - 3 X(9200) = 4 X(1637) - 3 X(9201) = 2 X(3268) - 3 X(9205) = 3 X(9195) - 2 X(11176)

X(14446) lies on these lines: {16, 9147}, {115, 125}, {3268, 6138}, {5466, 11602}, {9115, 13304}, {9195, 11176}

X(14446) = reflection of X(9201) in X(9200)
X(14446) = crossdifference of every pair of points on line {61, 110}
X(14446) = crosssum of X(16) and X(6137)
X(14446) = X(i)-isoconjugate of X(j) for these (i,j): {163, 11117}, {662, 2380}, {2154, 10409}
X(14446) = barycentric product X(i)*X(j) for these {i,j}: {523, 532}, {3268, 8014}
X(14446) = barycentric quotient X(i)/X(j) for these {i,j}: {16, 10409}, {512, 2380}, {523, 11117}, {532, 99}, {6138, 2981}, {8014, 476}

X(14447) =  TRIPOLAR CENTROID OF X(18)

Trilinears    Sin(A+Pi/3) Sin(B-C) (Cos(B-C)+2 Cos(A+Pi/3)) : :
Barycentrics   (b^2-c^2) (2 a^4-a^2 b^2-b^4-a^2 c^2+2 b^2 c^2-c^4-2 Sqrt(3) (2 a^2-b^2-c^2) S) : :
X(1447) = 4 X(1637) - 3 X(9200) = 2 X(1637) - 3 X(9201) = 2 X(3268) - 3 X(9204) = 3 X(9194) - 2 X(11176)

X(14447) lies on these lines: {15, 9147}, {115, 125}, {3268, 6137}, {5466, 11603}, {9117, 13305}, {9194, 11176}

X(14447) = reflection of X(9200) in X(9201)
X(14447) = crossdifference of every pair of points on line {62, 110, 2381, 3170, 3457, 6151}
X(14447) = crosssum of X(15) and X(6138)
X(14447) = X(i)-isoconjugate of X(j) for these (i,j): {163, 11118}, {662, 2381}, {2153, 10410}
X(14447) = barycentric product X(i)*X(j) for these {i,j}: {523, 533}, {3268, 8015}
X(14447) = barycentric quotient X(i)/X(j) for these {i,j}: {15, 10410}, {512, 2381}, {523, 11118}, {533, 99}, {6137, 6151}, {8015, 476}

X(14448) =  MIDPOINT OF X(7722) AND X(7731)

Barycentrics    a^2 (2 a^12 (b^2+c^2)-4 a^10 (2 b^4+b^2 c^2+2 c^4)-(b^2-c^2)^4 (2 b^6+5 b^4 c^2+5 b^2 c^4+2 c^6)+2 a^8 (5 b^6+2 b^4 c^2+2 b^2 c^4+5 c^6)-a^4 (b^2-c^2)^2 (10 b^6+b^4 c^2+b^2 c^4+10 c^6)+a^6 (-11 b^6 c^2+10 b^4 c^4-11 b^2 c^6)+a^2 (8 b^12-13 b^10 c^2+10 b^6 c^6-13 b^2 c^10+8 c^12)) : :
X(14448) = 9*X(51)-8*X(15465) = 3*X(51)-2*X(15738) = 3*X(113)-2*X(5876) = 3*X(125)-4*X(389) = 4*X(185)-3*X(17853) = 2*X(389)-3*X(1986) = 3*X(568)-2*X(36253) = X(6241)-3*X(7722) = X(6241)+3*X(7731) = 4*X(6241)-3*X(17856) = 4*X(7722)-X(17856) = 4*X(7731)+X(17856) = X(10990)-4*X(13148) = 2*X(10990)-3*X(17853) = 2*X(11381)-3*X(13202) = X(11381)-3*X(13417) = 8*X(13148)-3*X(17853) = 3*X(13202)-4*X(16105) = 3*X(13417)-2*X(16105) = 4*X(15465)-3*X(15738)

See Antreas Hatzipolakis and Angel Montesdeoca, Hyacinthos 26570.

X(14448) lies on these lines: {4,11564}, {51,15465}, {52,3627}, {74,578}, {113,5876}, {125,389}, {185,1205}, {399,15085}, {541,12897}, {542,5889}, {568,36253}, {576,10752}, {1112,21650}, {1154,30714}, {1192,17847}, {1495,5609}, {2777,6241}, {2904,13293}, {2935,34469}, {3043,10274}, {5562,5642}, {5890,20417}, {5907,12824}, {5972,11444}, {6102,16003}, {6243,17702}, {6293,36201}, {6723,15028}, {7503,34155}, {7687,9781}, {7723,11557}, {7728,9927}, {9707,13289}, {9730,38729}, {9786,15106}, {10117,12165}, {10620,36749}, {10625,11562}, {10721,34786}, {11002,15044}, {11425,17835}, {11426,19457}, {11561,32142}, {11591,38795}, {11799,13754}, {11807,12292}, {12111,38791}, {12358,16223}, {13201,37853}, {14049,21660}, {14094,26883}, {14531,14984}, {14708,38727}, {15026,23515}, {16176,37196}, {16219,35477}, {16534,18436}, {16880,18394}, {20397,37481}, {32235,37489}

X(14448) = midpoint of X(7722) and X(7731)
X(14448) = reflection of X(i) in X(j) for these (i,j): (125, 1986), (185, 13148), (5562, 25711), (7723, 11557), (10990, 185), (11381, 16105), (12111, 38791), (12219, 5972), (12281, 7687), (12292, 11807), (13201, 37853), (13202, 13417), (16003, 6102), (16163, 11562), (18436, 16534), (21650, 1112)
X(14448) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (185, 10990, 17853), (5562, 25711, 5642), (7723, 11557, 36518), (11381, 13417, 16105), (11381, 16105, 13202), (17835, 19504, 32607)

X(14449) =  MIDPOINT OF X(5) AND X(6243)

Barycentrics    a^2 (2 a^6 (b^2+c^2)-6 a^4 (b^4+b^2 c^2+c^4)-(b^2-c^2)^2 (2 b^4-b^2 c^2+2 c^4)+a^2 (6 b^6-b^4 c^2-b^2 c^4+6 c^6)) : :
X(14449) = X(4)-9*X(16981) = X(5)-3*X(3060) = 5*X(5)-7*X(9781) = 3*X(5)-X(11412) = 7*X(5)-5*X(11444) = 2*X(5)-3*X(13451) = 5*X(5)-3*X(23039) = 3*X(3060)+X(6243) = 15*X(3060)-7*X(9781) = 9*X(3060)-X(11412) = 5*X(3060)-X(23039) = 5*X(6243)+7*X(9781) = 3*X(6243)+X(11412) = 7*X(6243)+5*X(11444) = 2*X(6243)+3*X(13451) = 5*X(6243)+3*X(23039) = 14*X(9781)-15*X(13451) = 7*X(9781)-3*X(23039) = 7*X(11412)-15*X(11444) = 2*X(11412)-9*X(13451) = 5*X(11412)-9*X(23039)

See Antreas Hatzipolakis and Angel Montesdeoca, Hyacinthos 26570.

X(14449) lies on these lines: {3,16881}, {4,15110}, {5,3060}, {6,7525}, {23,195}, {26,1351}, {30,52}, {49,37936}, {51,3628}, {140,143}, {323,13621}, {389,548}, {397,36981}, {398,36979}, {428,6152}, {546,1154}, {547,1216}, {549,3567}, {550,568}, {576,7555}, {631,13321}, {632,2979}, {1112,10272}, {1147,37517}, {1199,13564}, {1614,37947}, {1656,11002}, {1658,36747}, {1993,37440}, {1994,2937}, {3530,5946}, {3627,5889}, {3819,32205}, {3845,18436}, {3850,5562}, {3853,13754}, {3856,15060}, {3858,11459}, {3861,5876}, {3917,15026}, {4888,13340}, {5066,10110}, {5133,21230}, {5663,13598}, {5752,7508}, {5890,15704}, {5891,12811}, {5901,31757}, {5943,32142}, {6403,7715}, {6515,18356}, {6676,8254}, {7502,36749}, {7512,14627}, {7516,9777}, {7530,12160}, {7553,32358}, {7575,34148}, {7999,15699}, {8703,37481}, {9704,11004}, {9729,34200}, {9730,33923}, {9820,10096}, {10024,20424}, {10113,18555}, {10170,18874}, {10282,12105}, {10540,15801}, {11250,37489}, {11477,12106}, {11539,15024}, {11663,15534}, {11793,12812}, {11819,32423}, {12006,12100}, {12088,15087}, {12102,12162}, {12103,13630}, {12111,15687}, {12134,32196}, {12161,17714}, {12290,33699}, {12307,35500}, {12325,37349}, {12606,30531}, {13352,15331}, {13367,34149}, {13383,15806}, {13417,16655}, {13431,24981}, {13562,34380}, {14865,32608}, {14891,16226}, {15038,37126}, {15047,15246}, {15056,38071}, {15067,35018}, {15646,37495}, {15761,31802}, {16657,32352}, {17712,19924}, {18128,29317}, {18282,32269}, {18377,31815}, {18569,18918}, {22352,36153}, {23061,38848}, {23292,34577}, {28178,31728}, {28186,31732}, {31670,32140}

X(14449) = midpoint of X(i) and X(j) for these {i,j}: {5, 6243}, {52, 10263}, {143, 13421}, {3627, 5889}, {5876, 14531}, {7553, 32358}
X(14449) = reflection of X(i) in X(j) for these (i,j): (3, 16881), (140, 143), (546, 5446), (547, 21849), (548, 389), (1216, 10095), (5446, 16982), (5562, 3850), (5876, 3861), (5901, 31757), (6101, 3628), (10272, 1112), (10625, 3530), (10627, 5462), (11591, 10110), (12103, 13630), (12162, 12102), (12606, 30531), (13451, 3060), (13630, 16625), (15644, 12006), (31834, 546)
X(14449) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (51, 6101, 3628), (52, 21969, 10263), (143, 10627, 5462), (1216, 10095, 547), (1216, 21849, 10095), (3060, 6243, 5), (3567, 37484, 549), (3917, 15026, 16239), (5447, 13363, 140), (5462, 10627, 140), (5946, 10625, 3530), (9781, 23039, 5), (10110, 11591, 5066), (11793, 13364, 12812), (12006, 15644, 12100), (12161, 33586, 17714)

X(14450) =  X(7)X(21)∩X(8)X(79)

Barycentrics    a^4+2 a^3 b-2 a b^3-b^4+2 a^3 c+a^2 b c+a b^2 c+a b c^2+2 b^2 c^2-2 a c^3-c^4 : :
X(14450) = X(8) - 4 X(79) = 4 X(21) - 5 X(3616) = 5 X(3616) - 2 X(3648) = 5 X(3616) - 8 X(3649) = X(3648) - 4 X(3649) = 7 X(3622) - 6 X(5426) = 8 X(3647) - 11 X(5550) = 4 X(5499) - 3 X(5657) = 4 X(3651) - 3 X(9778) = 8 X(6841) - 9 X(9779) = 8 X(442) - 7 X(9780) = 3 X(2) - 4 X(11263) = 15 X(3616) - 16 X(11281) = 3 X(3648) - 8 X(11281) = 3 X(21) - 4 X(11281) = 3 X(3649) - 2 X(11281) = 7 X(9780) - 16 X(11544) = 7 X(9780) - 4 X(11684) = 4 X(11544) - X(11684) = 3 X(5603) - 2 X(13743)

See Antreas Hatzipolakis, César Lozada, and Peter Moses, Hyacinthos 26579 and Hyacinthos 26591.

X(14450) lies on these lines: {1,5180}, {2,191}, {4,2771}, {5,13465}, {7,21}, {8,79}, {10,11552}, {30,944}, {65,5080}, {78,4312}, {80,4757}, {145,9802}, {149,3874}, {320,3702}, {329,442}, {404,11246}, {499,1749}, {942,5057}, {946,7701}, {1046,3120}, {1621,6147}, {1836,3868}, {2094,3652}, {2478,8261}, {3218,12047}, {3219,12609}, {3255,5558}, {3303,13995}, {3337,11813}, {3585,4084}, {3622,5426}, {3647,5550}, {3650,6675}, {3651,5758}, {3811,4338}, {3873,12699}, {3876,5880}, {3877,10404}, {3889,12701}, {4018,5086}, {4193,5221}, {4292,4511}, {4293,10052}, {4654,5250}, {4973,5443}, {5046,5902}, {5330,5434}, {5499,5657}, {5528,10123}, {5535,6960}, {5556,6598}, {5603,13743}, {5693,6839}, {5694,6901}, {5770,6841}, {5884,6840}, {5885,6902}, {6175,11236}, {6224,7354}, {6763,12535}, {9579,12866}, {9785,10543}, {9799,9812}, {9965,10527}, {10122,10580}, {10578,11508}, {12615,12937}, {12773,13126}

X(14450) = reflection of X(i) in X(j) for these {i,j}: {8,2475}, {21,3649}, {191,11263}, {442,11544}, {2475,79}, {3648,21}, {3650,6675}, {7701,946}, {10266,12913}, {11684,442}, {12535,13089}, {12937,12615}, {13465,5}
X(14450) = anticomplement of X(191)
X(14450) = X(54)-of-2nd-Conway-triangle
X(14450) = 2nd-Conway-isogonal conjugate of X(4)
X(14450) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {267, 8}, {502, 1330}, {1029, 69}, {3444, 2}
X(14450) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (191, 11263, 2), (2895, 4647, 8), (4295, 5905,8)

X(14451) =  ANTIGONAL CONJUGATE OF X(8487)

Barycentrics    (2*SA-3*R^2)*(S^2-9*R^2*SB+3*S B^2)*(S^2-9*R^2*SC+3*SC^2) : :

See Antreas Hatzipolakis and César Lozada, Hyacinthos 26592.

X(14451) lies on the cubic K060 and these lines: {30, 146}, {265, 1117}, {2201, 6365}, {6761, 11584}, {10272, 14354}

X(14451) = antigonal conjugate of X(8487)
X(14451) = perspector of ABC and 1st isogonal triangle of X(399)

X(14452) =  ANTIGONAL CONJUGATE OF X(7329)

Barycentrics    (a^3+(b+c)*a^2-(b^2-b*c+c^2)*a -(b^2-c^2)*(b-c))/((a^2-b^2+b* c-c^2)*(a^3+(b+c)*a^2-(b^2+b* c+c^2)*a-(b^2-c^2)*(b-c))) : :

See Antreas Hatzipolakis and César Lozada, Hyacinthos 26592.

X(14452) lies on the cubic K060 and these lines: {30, 80}, {79, 1117}

X(14452) = antigonal conjugate of X(7329)
X(14452) = perspector of ABC and 1st isogonal triangle of X(484)

X(14453)  =  (name pending)

Barycentrics    2 a^7-a^6 b-2 a^5 b^2+3 a^4 b^3-a^2 b^5-b^7-a^6 c+a^4 b^2 c-a^2 b^4 c+b^6 c-2 a^5 c^2+a^4 b c^2+4 a^3 b^2 c^2+2 a^2 b^3 c^2+3 b^5 c^2+3 a^4 c^3+2 a^2 b^2 c^3-3 b^4 c^3-a^2 b c^4-3 b^3 c^4-a^2 c^5+3 b^2 c^5+b c^6-c^7 : :

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 26590.

X(14453) lies on these lines: {1,14455}, {517,14454}, {17647,24269}, {23537,23850}

X(14454)  =  (name pending)

Barycentrics    a (a-b-c) (a^7 b-a^6 b^2-3 a^5 b^3+3 a^4 b^4+3 a^3 b^5-3 a^2 b^6-a b^7+b^8+a^7 c-a^5 b^2 c-2 a^4 b^3 c-a^3 b^4 c+4 a^2 b^5 c+a b^6 c-2 b^7 c-a^6 c^2-a^5 b c^2-2 a^4 b^2 c^2-2 a^3 b^3 c^2-a^2 b^4 c^2-a b^5 c^2-3 a^5 c^3-2 a^4 b c^3-2 a^3 b^2 c^3-8 a^2 b^3 c^3+a b^4 c^3+2 b^5 c^3+3 a^4 c^4-a^3 b c^4-a^2 b^2 c^4+a b^3 c^4-2 b^4 c^4+3 a^3 c^5+4 a^2 b c^5-a b^2 c^5+2 b^3 c^5-3 a^2 c^6+a b c^6-a c^7-2 b c^7+c^8) : :

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 26590.

X(14454) lies on these lines: {10,912}, {72,11507}, {78,1858}, {960,5248}, {5123,8261}, {6796,9943}

X(14455)  =  (name pending)

Barycentrics    a^6-a^4 b^2+a^2 b^4-b^6+2 a^4 b c+2 b^5 c-a^4 c^2+2 a^2 b^2 c^2+b^4 c^2-4 b^3 c^3+a^2 c^4+b^2 c^4+2 b c^5-c^6 : :
X(14455) = (R (3 r^2-s^2))/(2 r (r+R)^2) X(181) + X(5905)

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 26590.

X(14455) lies on these lines: {181,5905}, {197,3782}, {12588,14213}

X(14456)  =  (name pending)

Barycentrics    a^2 (a^14-4 a^12 b^2+3 a^10 b^4+8 a^8 b^6-17 a^6 b^8+12 a^4 b^10-3 a^2 b^12-4 a^12 c^2+12 a^10 b^2 c^2-2 a^8 b^4 c^2-16 a^6 b^6 c^2+8 a^4 b^8 c^2+4 a^2 b^10 c^2-2 b^12 c^2+3 a^10 c^4-2 a^8 b^2 c^4-6 a^6 b^4 c^4-20 a^4 b^6 c^4+19 a^2 b^8 c^4+6 b^10 c^4+8 a^8 c^6-16 a^6 b^2 c^6-20 a^4 b^4 c^6-40 a^2 b^6 c^6-4 b^8 c^6-17 a^6 c^8+8 a^4 b^2 c^8+19 a^2 b^4 c^8-4 b^6 c^8+12 a^4 c^10+4 a^2 b^2 c^10+6 b^4 c^10-3 a^2 c^12-2 b^2 c^12) : :

See Kostas Vittas and and Peter Moses, Hyacinthos 26595.

X(14456) lies on these lines: {647,11424}, {1971,9605}

X(14457)  =  PERSPECTOR OF THE YIU-HUTSON CONIC

Barycentrics    (a^8-2 a^6 b^2+2 a^4 b^4-2 a^2 b^6+b^8-2 a^6 c^2+6 a^4 b^2 c^2+6 a^2 b^4 c^2-2 b^6 c^2-6 a^2 b^2 c^4+2 a^2 c^6+2 b^2 c^6-c^8) (a^8-2 a^6 b^2+2 a^2 b^6-b^8-2 a^6 c^2+6 a^4 b^2 c^2-6 a^2 b^4 c^2+2 b^6 c^2+2 a^4 c^4+6 a^2 b^2 c^4-2 a^2 c^6-2 b^2 c^6+c^8) : :

See X(14390).

X(14457) lies on the Jerabek hyperbola, the cubic K520, and these lines: {3,12241}, {6,235}, {54,3542}, {64,1885}, {66,12294}, {68,5907}, {69,6816}, {389,4846}, {578,5504}, {895,3091}, {1112,11744}, {1352,6391}, {1619,6146}, {3147,3431}, {3426,14216}, {3448,11469}, {3527,12233}, {5286,13526}, {6526,14390}, {9815,11746}, {10574,11433}

X(14457) = X(i)-cross conjugate of X(j) for these (i,j): {5065, 2}, {6641, 2165}

X(14458)  =  ISOGONAL CONJUGATE OF X(3098)

Barycentrics    (2 a^4+2 a^2 b^2+2 b^4-a^2 c^2-b^2 c^2-c^4) (2 a^4-a^2 b^2-b^4+2 a^2 c^2-b^2 c^2+2 c^4) : :
X(14458) = 3 X(598) - 4 X(3845) = 9 X(262) - 8 X(9300) = 4 X(2) - 3 X(9774) = 2 X(2) - 3 X(10033) = 9 X(11167) - 8 X(13468)

Let A' be the orthocenter of BCX(76), and define B' and C' cyclically. Then X(14458) = X(2)-of-A'B'C'. (Randy Hutson, November 2, 2017)

X(14458) lies on the Kiepert hyperbola, the cubics K395 and K914, and these lines: {2,1495}, {3,10159}, {4,5007}, {17,1080}, {18,383}, {30,76}, {83,381}, {262,1503}, {275,5064}, {376,7800}, {428,2052}, {459,7714}, {512,2394}, {542,1916}, {598,3845}, {671,3830}, {804,14223}, {1513,7607}, {2782,10290}, {2996,3543}, {3399,11257}, {3424,9753}, {3524,7822}, {3534,10302}, {3545,7834}, {3590,7374}, {3591,7000}, {3839,5395}, {3849,5485}, {5055,7943}, {5306,9993}, {5503,9766}, {5989,6054}, {6039,6177}, {6040,6178}, {6811,10195}, {6813,10194}, {7470,7865}, {7608,13860}, {7710,10155}, {9302,9862}, {11058,14388}, {11167,13468}, {11177,11606}, {13748,14231}, {13749,14245}, {14230,14240}, {14233,14236}
X(14458) = reflection of X(9774) in X(10033)
X(14458) = isogonal conjugate of X(3098)
X(14458) = isotomic conjugate of X(7788)
X(14458) = X(i)-cross conjugate of X(j) for these (i,j): {5306, 2}, {9993, 262}
X(14458) = trilinear pole of line {523, 14398}
X(14458) = Cundy-Parry Phi transform of X(10159)
X(14458) = Cundy-Parry Psi transform of X(5007)
X(14458) = polar conjugate of X(11331)
X(14458) = X(i)-vertex conjugate of X(j) for these (i,j): {3, 262}, {54, 5481}, {262, 3}, {3425, 7607}, {5481, 54}, {7607, 3425}
X(14458) = isoconjugate of X(j) and X(j) for these (i,j): {1, 3098}, {31, 7788}, {48, 11331}, {662, 9210}
X(14458) = barycentric product X(i)* X(j) for these {i,j}: {6, 14387}, {512, 9211}
X(14458) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 7788}, {4, 11331}, {6, 3098}, {512, 9210}, {9211, 670}, {14387, 76}, {14398, 9411}

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Koutras-Hatzipolakis-Moses points: X(14459)-X(14478)

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Following problem 1165a in Stathis Koutras's posting to Romantics of Geometry, Antreas Hatzipolakis posed the following in Hyacinthos 26601, September 12, 2017:

Let P be a point in the plane of a triangle ABC. Let
LAC = line through A parallel to line CP, and define LBA and LCB cyclically
A' = BC∩LAC, and define B' and C' cyclically
MA = midpoint of A and A', and define MB and MC cyclically.
A* = AMC∩B'C', and define B* and C* cyclically.
The points A*, B*, C* are collinear (Koutras).

LAB = line through A parallel to line BP, and define LBC and LCA cyclically
A'' = BC∩LAB, and define B'' and C'' cyclically
NA = midpoint of A and A'', and define NB and NC cyclically.
A** = ANB∩B''C'', and define B** and C** cyclically.
The points A**, B**, C** are collinear. (Hatzipolakis).

What can be said about the point of intersection of the lines A*B*C* and A**B**C**? (Hatzipolakis)

Peter Moses responds as follows. Write P = p : q : r (barycentrics). The point of intersection, here denoted by KHM(P), lies on the line PX(2) and is given by

KHM(P) = p3 - q3 - r3 + 3p2(q + r) - 2qr(q + r) - pqr : :

and by the following combo:

KHM(P) = 3 (p3 + q3 + r3 + 2 (q r (q + r)+ r p (r + p) + p q (p + q) + 3 p q r))*X(2) - (p + q + r) (2 (p2 + q2 + r2) + 5 (q r + r p + p q))*P

The appearance of (i,j) in the following list means that KHM(X(i)) = X(j): (1,14459), (3,14460), (4,14461), (5,14462), (6,14463), (8,5212), (10,14464).

The mapping P → KHM(P) is a two-to-one mapping in the sense that if

P' = 3 p^4-6 p^3 q+2 p^2 q^2+8 p q^3-4 q^4-6 p^3 r+14 p^2 q r-14 p q^2 r+q^3 r+2 p^2 r^2-14 p q r^2+9 q^2 r^2+8 p r^3+q r^3-4 r^4 : : ,

then KHM(P') = KHM(P). An example of a pair (P,P') is (X(1),X(14478)). (Peter Moses, September 13, 2017).

The locus of a point P such that KHM(P) = P is the cubic, here named the KHM cubic, given by the following barycentric equation:

x3 + y3 + z3 + 2y2z + 2yz2 + 2x2z + 2xz2 + 2x2y + 2xy2 + 6xyz = 0.

If 1 ≤ i ≤ 15000, then X(i) does not lie on the KHM cubic. If P is a point then the line GP meets the KHM cubic in this point:

(2 p - q - r) ((p + q + r)^2 - 3 (q r + r p + p q) - 3 (q - r)^2 - (7 (27 p q r + 2 (p + q + r)^3 - 9 (p + q + r) ((q r + r p + p q)))^2)^(1/3)) : :

(Peter Moses, September 14, 2017)

X(14459)  =  KOUTRAS-HATZIPOLAKIS-MOSES POINT OF X(1)

Barycentrics    a^3+3 a^2 b-b^3+3 a^2 c-a b c-2 b^2 c-2 b c^2-c^3 : :

X(14459) lies on these lines: {1,2}, {320,4938}, {522,4813}, {740,5057}, {1155,4725}, {1776,2269}, {1914,3712}, {1962,4886}, {2895,4970}, {3936,4716}, {4009,4727}, {4819,5846}

X(14460)  =  KOUTRAS-HATZIPOLAKIS-MOSES POINT OF X(3)

Barycentrics    a^12-6 a^10 b^2+12 a^8 b^4-9 a^6 b^6+3 a^2 b^10-b^12-6 a^10 c^2+11 a^8 b^2 c^2-3 a^6 b^4 c^2+2 a^4 b^6 c^2-9 a^2 b^8 c^2+5 b^10 c^2+12 a^8 c^4-3 a^6 b^2 c^4-4 a^4 b^4 c^4+6 a^2 b^6 c^4-11 b^8 c^4-9 a^6 c^6+2 a^4 b^2 c^6+6 a^2 b^4 c^6+14 b^6 c^6-9 a^2 b^2 c^8-11 b^4 c^8+3 a^2 c^10+5 b^2 c^10-c^12 : :

X(14460) lies on this line: {2,3}

X(14461)  =  KOUTRAS-HATZIPOLAKIS-MOSES POINT OF X(4)

Barycentrics    5 a^10 b^2-15 a^8 b^4+14 a^6 b^6-2 a^4 b^8-3 a^2 b^10+b^12+5 a^10 c^2-4 a^8 b^2 c^2-2 a^6 b^4 c^2-8 a^4 b^6 c^2+13 a^2 b^8 c^2-4 b^10 c^2-15 a^8 c^4-2 a^6 b^2 c^4+20 a^4 b^4 c^4-10 a^2 b^6 c^4+7 b^8 c^4+14 a^6 c^6-8 a^4 b^2 c^6-10 a^2 b^4 c^6-8 b^6 c^6-2 a^4 c^8+13 a^2 b^2 c^8+7 b^4 c^8-3 a^2 c^10-4 b^2 c^10+c^12 : :

X(14461) lies on this line: {2,3}

X(14461) = crossdifference of every pair of points on line X(647)X(11424)

X(14462)  =  KOUTRAS-HATZIPOLAKIS-MOSES POINT OF X(5)

Barycentrics    6 a^12-28 a^10 b^2+48 a^8 b^4-33 a^6 b^6+a^4 b^8+9 a^2 b^10-3 b^12-28 a^10 c^2+54 a^8 b^2 c^2-17 a^6 b^4 c^2+2 a^4 b^6 c^2-29 a^2 b^8 c^2+18 b^10 c^2+48 a^8 c^4-17 a^6 b^2 c^4-6 a^4 b^4 c^4+20 a^2 b^6 c^4-45 b^8 c^4-33 a^6 c^6+2 a^4 b^2 c^6+20 a^2 b^4 c^6+60 b^6 c^6+a^4 c^8-29 a^2 b^2 c^8-45 b^4 c^8+9 a^2 c^10+18 b^2 c^10-3 c^12 : :

X(14462) lies on this line: {2,3}

X(14463)  =  KOUTRAS-HATZIPOLAKIS-MOSES POINT OF X(6)

Barycentrics    a^6+3 a^4 b^2-b^6+3 a^4 c^2-a^2 b^2 c^2-2 b^4 c^2-2 b^2 c^4-c^6 : :

X(14463) lies on these lines: {2,6}, {525,8665}

X(14464)  =  KOUTRAS-HATZIPOLAKIS-MOSES POINT OF X(10)

Barycentrics    6 a^3+10 a^2 b-3 b^3+10 a^2 c-2 a b c-9 b^2 c-9 b c^2-3 c^3 : :

X(14464) lies on this line: {1,2}

X(14465)  =  KOUTRAS-HATZIPOLAKIS-MOSES POINT OF X(31)

Barycentrics    a^9 + 3*a^6*b^3 - b^9 + 3*a^6*c^3 - a^3*b^3*c^3 - 2*b^6*c^3 - 2*b^3*c^6 - c^9 : :

X(14465) lies on this line: {2, 31}

X(14466)  =  KOUTRAS-HATZIPOLAKIS-MOSES POINT OF X(32)

Barycentrics    a^12 + 3*a^8*b^4 - b^12 + 3*a^8*c^4 - a^4*b^4*c^4 - 2*b^8*c^4 - 2*b^4*c^8 - c^12 : :

X(14466) lies on this line: {2, 32}

X(14467)  =  KOUTRAS-HATZIPOLAKIS-MOSES POINT OF X(69)

Barycentrics    5*a^4*b^2 - b^6 + 5*a^4*c^2 - 6*a^2*b^2*c^2 - b^4*c^2 - b^2*c^4 - c^6 : :

X(14465) lies on these lines: {2, 6}, {3005, 3566}, {5651, 7762}, {7793, 13394}

X(14468)  =  KOUTRAS-HATZIPOLAKIS-MOSES POINT OF X(141)

Barycentrics   6*a^6 + 10*a^4*b^2 - 3*b^6 + 10*a^4*c^2 - 2*a^2*b^2*c^2 - 9*b^4*c^2 - 9*b^2*c^4 - 3*c^6 : :

X(14468) lies on this line: {2, 6}

X(14469)  =  KOUTRAS-HATZIPOLAKIS-MOSES POINT OF X(145)

Barycentrics    6*a^3 - 13*a^2*b + 3*b^3 - 13*a^2*c + 20*a*b*c - 3*b^2*c - 3*b*c^2 + 3*c^3 : :

X(14469) lies on these lines: {1, 2}, {650, 4962}

X(14470)  =  KOUTRAS-HATZIPOLAKIS-MOSES POINT OF X(192)

Barycentrics    a*(a^2*b^3 + a^2*b^2*c + a^2*b*c^2 + 6*a*b^2*c^2 - 5*b^3*c^2 + a^2*c^3 - 5*b^2*c^3) : :

X(14470) lies on this line: {2, 37}

X(14471)  =  KOUTRAS-HATZIPOLAKIS-MOSES POINT OF X(193)

Barycentrics    6*a^6 - 13*a^4*b^2 + 3*b^6 - 13*a^4*c^2 + 20*a^2*b^2*c^2 - 3*b^4*c^2 - 3*b^2*c^4 + 3*c^6 : :

X(14471) lies on this line: {2, 6}

X(14472)  =  KOUTRAS-HATZIPOLAKIS-MOSES POINT OF X(194)

Barycentrics    a^2*(a^4*b^6 + a^4*b^4*c^2 + a^4*b^2*c^4 + 6*a^2*b^4*c^4 - 5*b^6*c^4 + a^4*c^6 - 5*b^4*c^6) : :

X(14472) lies on this line: {2, 39}

X(14473)  =  KOUTRAS-HATZIPOLAKIS-MOSES POINT OF X(315)

Barycentrics    5*a^8*b^4 - b^12 + 5*a^8*c^4 - 6*a^4*b^4*c^4 - b^8*c^4 - b^4*c^8 - c^12 : :

X(14473) lies on this line: {2, 32}

X(14474)  =  KOUTRAS-HATZIPOLAKIS-MOSES POINT OF X(513)

Barycentrics    a (b-c) (a^2 b^2-4 a^2 b c+2 a b^2 c+a^2 c^2+2 a b c^2-2 b^2 c^2) : :

X(14474) lies on these lines: on lines {2, 513}, {244, 665}, {667, 750}, {3572, 4893}, {4763, 14404}

X(14475)  =  KOUTRAS-HATZIPOLAKIS-MOSES POINT OF X(514)

Barycentrics    (b-c) (-2 a^2+2 a b+b^2+2 a c-4 b c+c^2) : :

X(14475) lies on these lines: {2,514}, {11,244}, {649,3306}, {812,14435}, {1635,4927}, {1644,14410}, {3667,9779}, {3676,5219}, {4120,4453}, {4382,7658}, {4444,4776}, {4688,4777}, {5550,5592}, {6006,6173}

X(14476)  =  KOUTRAS-HATZIPOLAKIS-MOSES POINT OF X(522)

Barycentrics    (a-b-c) (b-c) (2 a^4-2 a^3 b-3 a^2 b^2+4 a b^3-b^4-2 a^3 c+8 a^2 b c-4 a b^2 c-2 b^3 c-3 a^2 c^2-4 a b c^2+6 b^2 c^2+4 a c^3-2 b c^3-c^4) : :

X(14476) lies on these lines: {2,522}, {11,1146}, {514,7988}, {663,9817}, {3239,5231}

X(14477)  =  KOUTRAS-HATZIPOLAKIS-MOSES POINT OF X(527)

Barycentrics    (2 a^2-a b-b^2-a c+2 b c-c^2) (2 a^4-2 a^3 b-3 a^2 b^2+4 a b^3-b^4-2 a^3 c+8 a^2 b c-4 a b^2 c-2 b^3 c-3 a^2 c^2-4 a b c^2+6 b^2 c^2+4 a c^3-2 b c^3-c^4) : :

X(14477) lies on these lines: {2,7}, {1638,6174}, {10708,11219}

X(14478)  =  (name pending)

Barycentrics    3 a^4-6 a^3 b+2 a^2 b^2+8 a b^3-4 b^4-6 a^3 c+14 a^2 b c-14 a b^2 c+b^3 c+2 a^2 c^2-14 a b c^2+9 b^2 c^2+8 a c^3+b c^3-4 c^4 : :

There are two points X such that KHM(X) = X(14459); they are X(1) and X(14478). See the preamble just before X(14459).

X(14478) lies on theis line: {1,2}

X(14479)  =  ISOGONAL CONJUGATE OF X(11057)

Barycentrics    a^2 (2 a^4-a^2 b^2+2 b^4+2 a^2 c^2+2 b^2 c^2-4 c^4) (2 a^4+2 a^2 b^2-4 b^4-a^2 c^2+2 b^2 c^2+2 c^4) : :

X(14479) lies on the cubic K914 and these lines: {599,3098}, {11738,14388}

X(14479) = isogonal conjugate of X(11057)
X(14479) = isoconjugate of X(j) and X(j) for these (i,j): {1, 11057}, {75, 7712}
X(14479) = barycentric product X(6)*X(11058)
X(14479) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 11057}, {32, 7712}, {11058, 76}

X(14480)  =  ANTICOMPLEMENT OF X(6070)

Barycentrics    (a^2-b^2) (a^2-c^2) (a^8-a^6 b^2-2 a^4 b^4+3 a^2 b^6-b^8-a^6 c^2+5 a^4 b^2 c^2-3 a^2 b^4 c^2+4 b^6 c^2-2 a^4 c^4-3 a^2 b^2 c^4-6 b^4 c^4+3 a^2 c^6+4 b^2 c^6-c^8) : :
X(14480) = 5 X(476) - 8 X(3233) = 5 X(110) - 4 X(3233) = 4 X(5) - 3 X(5627) = 6 X(3233) - 5 X(7471) = 3 X(476) - 4 X(7471) = 3 X(110) - 2 X(7471) = 4 X(3154) - 3 X(9140)

X(14480) lies on the curve Q000aH3 and these lines: {2,6070}, {5,1117}, {30,14094}, {110,476}, {477,5663}, {520,11751}, {524,9158}, {648,7480}, {930,10420}, {1316,11422}, {2452,5640}, {3154,9140}, {3258,3448}, {4226,14366}, {6795,7998}

X(14480) = reflection of X(i) in X(j) for these {i,j}: {476, 110}, {3448, 3258}
X(14480) = anticomplement X(6070)

X(14481)  =  ISOGONAL CONJUGATE OF X(3356)

Barycentrics    a^2 (b^2 r^2 p^2 + c^2 p^2 q^2 - a^2 q^2 r^2) / (1 / (q SB) + 1 / (r SC) - 1 / (p SA)) : : , where p : q : r = X(20).

X(14481) lies on the cubic K002 and these lines: {2,3356}, {3,3637}, {4,3344}, {6,3349}, {57,3352}, {282,3342}

X(14481) = isogonal conjugate of X(3356)
X(14481) = Cundy-Parry Phi transform of X(3637)
X(14481) = Cundy-Parry Psi transform of X(3355)
X(14481) = X(31)-complementary conjugate of X(3349)
X(14481) = X(2)-Ceva conjugate of X(3349)
X(14481) = perspector of ABC and antipedal triangle of X(3637)
X(14481) = perspector of ABC and medial triangle of pedal triangle of X(3355)
X(14481) = complement of perspector of ABC and pedal triangle of X(2131)
X(14481) = perspector of circumconic centered at X(3349)
X(14481) = barycentric product X(2130)*X(14365)
X(14481) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 3356}, {2130, 14362}

X(14482)  =  CROSSSUM OF X(6) AND X(5646)

Barycentrics    3 a^4+12 a^2 b^2+b^4+12 a^2 c^2-2 b^2 c^2+c^4 : :

X(14482) lies on these lines: {4,9605}, {6,376}, {39,631}, {40,4253}, {112,5702}, {115,3545}, {148,8592}, {378,1033}, {551,3247}, {597,9741}, {1007,7827}, {1108,5069}, {1180,6353}, {1640,9168}, {3090,3815}, {3524,5024}, {3525,5305}, {3529,7737}, {3618,7757}, {3767,12815}, {3855,5254}, {5013,10299}, {6329,8716}, {6770,9112}, {6773,9113}, {7581,12257}, {7582,12256}, {7803,7909}, {7831,11008}, {9593,12245}, {9862,10753}

X(14482) = crosssum of X(6) and X(5646)
X(14482) = {X(5024),X(5304)}-harmonic conjugate of X(3524)

leftri

Nguyen-Moses images: X(14483)-X(14498)

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Let ABC be a triangle, let L be the line through B orthogonal to BC, let BA be any point on L, and let BC be the point such that BBABCC is a rectangle. Likewise, let CCBCAA be a rectangle with base CA and let AACABB be a rectangle with base AB. Let LA be the line through A perpendicular to BCCB, and define LB and LC cyclically. Starting with Nguyen Ngoc Giang's rectangles (as in the preamble just before X(14206)), Peter Moses (September 19, 2017) found that the lines LA, LB, LC concur.

Let U be the ratio of the height of the rectangle BBABCC to the base; that is, U = |BAB|/a. Define V and W cyclically. The point X' of concurrence is given by

X' = 1 / (-a^2 + b^2 + c^2 + 2 S U) : : (barycentrics)

If X is a point in the plane of ABC, then it has actual trilinear distances (possibly nonpositive) that are the heights of rectangles as in the above construction. Therefore, starting with X = x : y : z (barycentrics), we have U = kx/a2, where k = S/(x + y + z), and V = ky/b2 and W = kz/c2. Consequently,

X' = a^2 / (a^2 (-a^2 + b^2 + c^2) + 2 S^2 x (x + y + z)) : :

The point X' is here named the Nguyen-Moses image of X.

The appearance of (i,j) in the following list means that the X(j) = Nguyen-Moses image of X(i):

(1,3577), (2,3531), (3,4), (4,6), (5,14483), (6,14484), (39,14485), (20,3426), (23,265), (40,3062), (69, 14486), (140, 14487), (182,14488), (193,14489), (376,14489), (381,14491), (382,3431), (546,1173), (550,13603), (576,262), (575,14492), (1071,14493), (1181,8801), (1351,14494), (1352,14495), (1385,14496), (1482,14497), (1498,253), (1513,14498), (1657,11738), (2996,1351), (3091,3527), (3095,11170), (3146,3), (3529,64), (3627,54), (5076,13472), (5609,5627), (5881,2163), (6912,1243), (7464,11744), (7982,1), (7991,84), (9716,381), (10222,1389), (10594,14457), (11412,2980), (11477,2), (11541,3532), (12082,66), (12086,3521), (12088,6145), (12160,8797), (14094,523)

X(14483)  =  NGUYEN-MOSES IMAGE OF X(5)

Barycentrics    a^2*(a^4 - 5*a^2*b^2 + 4*b^4 - 2*a^2*c^2 - 5*b^2*c^2 + c^4)*(a^4 - 2*a^2*b^2 + b^4 - 5*a^2*c^2 - 5*b^2*c^2 + 4*c^4) : :
X(14483) = 2 X(3) - 5 X(5643)

X(14483) lies on the Jerabek hyperbola and these lines: {3, 5640}, {6, 14157}, {30, 13623}, {51, 74}, {54, 1495}, {64, 3567}, {67, 5480}, {68, 3832}, {69, 1568}, {72, 11278}, {248, 5008}, {265, 3845}, {403, 13622}, {879, 12073}, {895, 5097}, {1176, 13446}, {1511, 13482}, {1614, 13472}, {3426, 5890}, {3431, 11202}, {3519, 3850}, {3521, 3853}, {3527, 11456}, {3531, 9777}, {3543, 4846}, {3574, 13418}, {3581, 13451}, {4550, 11002}, {5318, 11138}, {5321, 11139}, {5504, 13595}, {5888, 13391}, {6000, 13603}, {10293, 10721}, {10982, 11464}, {11270, 11438}

X(14483) = isogonal conjugate of X(549)
X(14483) = X(12112)-cross conjugate of X(74)
X(14483) = X(i)-isoconjugate of X(j) for these (i,j): {1, 549}, {63, 6749}
X(14483) = X(i)-vertex conjugate of X(j) for these (i,j): {3, 13603}, {6, 54}
X(14483) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 549}, {25, 6749}

X(14484)  =  NGUYEN-MOSES IMAGE OF X(6)

Barycentrics    (a^4 + 6*a^2*b^2 + b^4 + 2*a^2*c^2 + 2*b^2*c^2 - 3*c^4)*(a^4 + 2*a^2*b^2 - 3*b^4 + 6*a^2*c^2 + 2*b^2*c^2 + c^4) : :

X(14484) lies on the Kiepert hyperbola and these lines: {2, 1350}, {4, 9605}, {6, 3424}, {10, 7407}, {20, 83}, {76, 3091}, {98, 5039}, {114, 5503}, {147, 671}, {226, 3677}, {275, 6995}, {381, 5485}, {427, 459}, {485, 6201}, {486, 6202}, {598, 3543}, {801, 7398}, {1297, 11348}, {1587, 14237}, {1588, 14232}, {2052, 7378}, {2996, 3832}, {3146, 3329}, {3314, 5068}, {3316, 6813}, {3317, 6811}, {3406, 10788}, {5032, 5984}, {5056, 10159}, {6036, 10153}, {6504, 7394}, {6776, 14458}, {7409, 8796}, {7519, 7578}, {7581, 14243}, {7582, 14228}, {7607, 9753}, {7608, 9993}, {7612, 9748}, {9740, 11167}, {10513, 10516}

X(14484) = isogonal conjugate of X(5085)
X(14484) = X(7736)-cross conjugate of X(2)
X(14484) = cevapoint of X(11) and X(2526)
X(14484) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 5085}

X(14485)  =  NGUYEN-MOSES IMAGE OF X(39)

Barycentrics    (a^6 - 5*a^4*b^2 - 5*a^2*b^4 + b^6 - 6*a^4*c^2 - 8*a^2*b^2*c^2 - 6*b^4*c^2 + 5*a^2*c^4 + 5*b^2*c^4)*(a^6 - 6*a^4*b^2 + 5*a^2*b^4 - 5*a^4*c^2 - 8*a^2*b^2*c^2 + 5*b^4*c^2 - 5*a^2*c^4 - 6*b^2*c^4 + c^6) : :

X(14483) lies on the Kiepert hyperbola and these lines: {2, 8722}, {76, 11477}, {262, 5024}, {381, 11167}, {671, 5480}, {5395, 12203}, {5503, 8724}, {7607, 12110}, {7612, 10788}

X(14486)  =  NGUYEN-MOSES IMAGE OF X(69)

Barycentrics    a^2*(a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*(a^4 - 2*a^2*b^2 + b^4 - 4*a^2*c^2 - 4*b^2*c^2 - c^4)*(a^4 - 4*a^2*b^2 - b^4 - 2*a^2*c^2 - 4*b^2*c^2 + c^4) : :

X(14486) lies on these lines: {4, 183}, {25, 182}, {95, 13860}, {1598, 2207}, {5198, 14248}, {6524, 6995}

X(14486) = isogonal conjugate of X(10519)
X(14486) = X(i)-isoconjugate of X(j) for these (i,j): {1, 10519}, {63, 7736}
X(14486) = trilinear pole of line {2489, 3288}
X(14486) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 10519}, {25, 7736}

X(14487)  =  NGUYEN-MOSES IMAGE OF X(140)

Barycentrics    a^2*(a^4 - 11*a^2*b^2 + 10*b^4 - 2*a^2*c^2 - 11*b^2*c^2 + c^4)*(a^4 - 2*a^2*b^2 + b^4 - 11*a^2*c^2 - 11*b^2*c^2 + 10*c^4) : :

X(14487) lies on the Jerabek hyperbola and these lines: {3, 10545}, {51, 13603}, {1173, 12112}, {5900, 7687}

X(14487) = isogonal conjugate of X(12100)
X(14487) = X(1173)-vertex conjugate of X(3431)
X(14487) = barycentric quotient X(6)/X(1210)

X(14488)  =  NGUYEN-MOSES IMAGE OF X(182)

Barycentrics    (2*a^4 + 6*a^2*b^2 + 2*b^4 + a^2*c^2 + b^2*c^2 - 3*c^4)*(2*a^4 + a^2*b^2 - 3*b^4 + 6*a^2*c^2 + b^2*c^2 + 2*c^4) : :

X(14488) lies on the Kiepert hyperbola and these lines: {4, 5041}, {76, 546}, {83, 382}, {381, 10302}, {671, 14269}, {1513, 11669}, {3529, 7804}, {3851, 7934}, {3855, 7849}, {5480, 14458}, {7607, 9993}

X(14488) = X(3425)-vertex conjugate of X(11669)

X(14489)  =  NGUYEN-MOSES IMAGE OF X(193)

Barycentrics    a^2*(a^6 - a^4*b^2 - a^2*b^4 + b^6 - 3*a^4*c^2 + 6*a^2*b^2*c^2 - 3*b^4*c^2 + 7*a^2*c^4 + 7*b^2*c^4 - 5*c^6)*(a^6 - 3*a^4*b^2 + 7*a^2*b^4 - 5*b^6 - a^4*c^2 + 6*a^2*b^2*c^2 + 7*b^4*c^2 - a^2*c^4 - 3*b^2*c^4 + c^6) : :
Barycentrics    (SB + SC)/(S^2(2 SA - SW) - SA SW (SB + SC)) : :

X(14489) lies on these lines: {3, 3199}, {5, 1007}, {25, 97}, {51, 394}, {276, 1093}, {1656, 14376}, {2967, 11284}, {3095, 11484}, {5093, 10314}

X(14489) = cevapoint of X(3) and X(5093)

X(14490)  =  NGUYEN-MOSES IMAGE OF X(376)

Barycentrics    a^2*(a^4 - 2*a^2*b^2 + b^4 + 10*a^2*c^2 + 10*b^2*c^2 - 11*c^4)*(a^4 + 10*a^2*b^2 - 11*b^4 - 2*a^2*c^2 + 10*b^2*c^2 + c^4) : :
X(14490) = 4 X(3) - 5 X(5646)

X(14490) lies on the Jerabek hyhperbola and these lines: {3, 5646}, {67, 13202}, {68, 3853}, {69, 3543}, {154, 3431}, {248, 9412}, {381, 13623}, {895, 5102}, {1439, 7274}, {3527, 13474}, {3531, 6000}, {3845, 4846}, {5505, 12133}, {6413, 6437}, {6414, 6438}, {7687, 10293}, {10605, 11738}, {11456, 13472}

X(14490) = isogonal conjugate of X(10304)
X(14490) = X(3)-vertex conjugate of X(3531)
X(14490) = X(i)-isoconjugate of X(j) for these (i,j): {1, 10304}, {63, 5702}
X(14490) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 10304}, {25, 5702}

X(14491)  =  NGUYEN-MOSES IMAGE OF X(381)

Barycentrics    a^2*(2*a^4 - 7*a^2*b^2 + 5*b^4 - 4*a^2*c^2 - 7*b^2*c^2 + 2*c^4)*(2*a^4 - 4*a^2*b^2 + 2*b^4 - 7*a^2*c^2 - 7*b^2*c^2 + 5*c^4) : :
X(14491) = 2 X(3) - 7 X(5645)

From Angel Montesdeoca, February 28, 2020:

Let ℰa be the ellipse with major axis the segment BC and length of the minor axis a/Sqrt[3]. Let La be the polar of A wrt ℰa, and define Lb and Lc cyclically. Let A' = Lb∩Lc, B' = Lc∩La, C' = La∩Lb. The lines AA', BB', CC' concur in X(14491). A barycentric equation for the ellipse ℰa:

2 a4 x2+a4 x y+a4 x z+2 a4 y z-3 a2 b2 x2-a2 b2 x y+a2 b2 x z-3 a2 c2 x2+a2 c2 x y-a2 c2 x z+b4 x2-2 b2 c2 x2+c4 x2=0.

X(14491) lies on the Jerabek hyperbola and these lines: {3, 5645}, {51, 3431}, {68, 3855}, {69, 5071}, {265, 3839}, {389, 13452}, {3426, 9777}, {3519, 5068}, {3531, 12112}, {3532, 10982}, {3567, 11270}, {5334, 11139}, {5335, 11138}, {5890, 11738}, {7687, 11564}, {7712, 13451}, {9781, 13472}

X(14491) = isogonal conjugate of X(5054)
X(14491) = X(3431)-vertex conjugate of X(3531)
X(14491) = barycentric quotient X(6)/X(5054)

X(14492)  =  NGUYEN-MOSES IMAGE OF X(575)

Barycentrics    (a^4 + 4*a^2*b^2 + b^4 + a^2*c^2 + b^2*c^2 - 2*c^4)*(a^4 + a^2*b^2 - 2*b^4 + 4*a^2*c^2 + b^2*c^2 + c^4) : :
X(14492) = 8 X(5066) - 3 X(10302)

X(14492) lies on the Kiepert hyperbola and these lines: {2, 3098}, {4, 7739}, {5, 10159}, {6, 14458}, {17, 383}, {18, 1080}, {30, 83}, {76, 381}, {98, 5306}, {115, 9302}, {275, 428}, {542, 11606}, {598, 3830}, {626, 3545}, {671, 3845}, {826, 2394}, {1513, 7608}, {1916, 6054}, {2052, 5064}, {2996, 3839}, {3406, 12110}, {3407, 5476}, {3524, 7889}, {3543, 5395}, {3590, 7000}, {3591, 7374}, {3818, 7837}, {5055, 7944}, {5066, 10302}, {5149, 12117}, {5478, 11603}, {5479, 11602}, {5969, 9765}, {6201, 14244}, {6202, 14229}, {6811, 10194}, {6813, 10195}, {7607, 13860}, {7612, 9753}, {8667, 11167}, {9479, 14223}

X(14492) = reflection of X(9302) in X(115)
X(14492) = isogonal conjugate of X(5092)
X(14492) = antigonal image of X(9302)
X(14492) = X(9300)-cross conjugate of X(2)
X(14492) = X(3425)-vertex conjugate of X(7608)
X(14492) = cevapoint of X(2) and X(7837)
X(14492) = trilinear pole of line {523, 9210}
X(14492) = barycentric quotient X(6)/X(5092)
X(14492) = center of inverse similitude of Ehrmann mid-triangle and 1st anti-Brocard triangle

X(14493)  =  NGUYEN-MOSES IMAGE OF X(1071)

Barycentrics    a*(a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*(a^3 - a^2*b - a*b^2 + b^3 - 2*a^2*c + 4*a*b*c - 2*b^2*c + a*c^2 + b*c^2)*(a^3 - 2*a^2*b + a*b^2 - a^2*c + 4*a*b*c + b^2*c - a*c^2 - 2*b*c^2 + c^3) : :

X(14493) lies on these lines: {19, 7994}, {28, 1902}, {34, 1697}, {1119, 3672}

X(14493) = isogonal conjugate of X(10167)
X(14493) = cevapoint of X(19) and X(7071)
X(14493) = X(i)-isoconjugate of X(j) for these (i,j): {1, 10167}, {3, 11019}, {77, 14100}, {348, 1200}
X(14493) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 10167}, {19, 11019}, {607, 14100}, {2212, 1200}

X(14494)  =  NGUYEN-MOSES IMAGE OF X(1351)

Barycentrics    (a^4 - 4*a^2*b^2 + 3*b^4 - 6*a^2*c^2 - 4*b^2*c^2 + c^4)*(a^4 - 6*a^2*b^2 + b^4 - 4*a^2*c^2 - 4*b^2*c^2 + 3*c^4) : :
Barycentrics    (tan A)/(2 tan A + cot ω) : :

Let O* be the orthoptic circle of the Steiner inellipse. Let A1 and A2 be the intersections of O* and line BC, and define B1, B2, C1, C2 cyclically. Let A' be the circumcenter of X(2)A1A2, and define B' and C' cyclically. The lines AA', BB', CC' concur in X(14494). (Randy Hutson, November 2, 2017)

X(14494) lies on the Kiepert hyperbola and these lines: {2, 1351}, {3, 5395}, {4, 3815}, {5, 2996}, {6, 7612}, {76, 1007}, {83, 631}, {98, 5034}, {114, 671}, {275, 6353}, {376, 598}, {427, 8796}, {1131, 6813}, {1132, 6811}, {2052, 8889}, {3055, 9752}, {3424, 13860}, {5067, 7778}, {5071, 5485}, {6504, 7392}, {6721, 8781}, {6997, 13579}, {7391, 11538}, {7394, 13585}, {7493, 7578}, {7607, 7735}, {7608, 9753}, {9744, 14458}, {9770, 11167}, {11163, 11172}, {11272, 14064}

X(14494) = isogonal conjugate of X(5050)
X(14494) = isotomic conjugate of X(34229)
X(14494) = barycentric quotient X(6)/X(5050)

X(14495)  =  NGUYEN-MOSES IMAGE OF X(1352)

Barycentrics    a^2*(a^6 - 3*a^4*b^2 + a^2*b^4 + b^6 - a^4*c^2 - 6*a^2*b^2*c^2 + b^4*c^2 - a^2*c^4 - 3*b^2*c^4 + c^6)*(a^6 - a^4*b^2 - a^2*b^4 + b^6 - 3*a^4*c^2 - 6*a^2*b^2*c^2 - 3*b^4*c^2 + a^2*c^4 + b^2*c^4 + c^6) : :
Barycentrics    (SB + SC)/(S^2 (SA + SW) - SA SW (2 SA + SB + SC - 2 SW)) : :

X(14495) lies on these lines: {5, 183}, {22, 51}, {95, 262}, {132, 11605}, {2980, 9755}, {3199, 5007}, {3425, 9969}, {5188, 7509}, {6249, 7566}

X(14495) = trilinear pole of line {2485, 3288}

X(14496)  =  NGUYEN-MOSES IMAGE OF X(1385)

Barycentrics    a*(a^3 - 6*a^2*b - a*b^2 + 6*b^3 - a^2*c + 2*a*b*c - b^2*c - a*c^2 - 6*b*c^2 + c^3)*(a^3 - a^2*b - a*b^2 + b^3 - 6*a^2*c + 2*a*b*c - 6*b^2*c - a*c^2 - b*c^2 + 6*c^3) : :

X(14496) lies on the Feuerbach hyperbola and these lines: {8, 546}, {495, 7320}, {946, 13606}, {1000, 10590}, {1156, 6797}, {7686, 10308}

X(14497)  =  NGUYEN-MOSES IMAGE OF X(1482)

Barycentrics    a*(2*a^3 - 3*a^2*b - 2*a*b^2 + 3*b^3 - 2*a^2*c + 4*a*b*c - 2*b^2*c - 2*a*c^2 - 3*b*c^2 + 2*c^3)*(2*a^3 - 2*a^2*b - 2*a*b^2 + 2*b^3 - 3*a^2*c + 4*a*b*c - 3*b^2*c - 2*a*c^2 - 2*b*c^2 + 3*c^3) : :

X(14497) lies on these lines: {1, 6942}, {4, 11011}, {7, 7967}, {8, 3090}, {21, 1482}, {79, 944}, {80, 5603}, {104, 2099}, {145, 6867}, {498, 5559}, {515, 5561}, {517, 2320}, {943, 2098}, {946, 5560}, {962, 10266}, {1000, 5048}, {1156, 10698}, {1320, 6911}, {3065, 13253}, {3241, 11604}, {5555, 10805}, {5734, 10526}

X(14497) = isogonal conjugate of X(10246)
X(14497) = X(3417)-vertex conjugate of X(3577)
X(14497) = barycentric quotient X(6)/X(10246)

X(14498)  =  NGUYEN-MOSES IMAGE OF X(1513)

Barycentrics    a^2*(a^4 - 3*a^2*b^2 + 4*b^4 - 3*b^2*c^2 + c^4)*(a^4 + b^4 - 3*a^2*c^2 - 3*b^2*c^2 + 4*c^4) : :

X(14498) lies on the Jerabek hyperbola and these lines: {3, 3124}, {6, 2971}, {69, 115}, {248, 2031}, {895, 1570}

X(14498) = isogonal conjugate of X(35297)

X(14499)  =  MOSES-STEINER IMAGE OF X(1114)

Barycentrics    SA (S^2-3 SB SC) ((1+J) S^2-(3-J) SB SC) : :

Points X(14499)-X(14507) are Moses Steiner images, on the Steiner deltoid, sometimes coded as Q000H3 or simply H3; see Q000. For the definition of Moses-Steiner iamge, see the preamble just before X(6070). Points on the Steiner deltoid include X(1553), X(6070)-X(6077), and X(14498)-X(14515).

X(14499) lies on the Steiner deltoid and these lines: {4,13414}, {30,113}, {125,1313}, {542,8116}, {1113,5972}, {1114,2777}

X(14499) = reflection of X(i) in X(j) for these {i,j}: {125, 1313}, {1113, 5972}
X(14499) = barycentric product X(i)X(j) for these {i,j}: {1313, 11064}

X(14500)  =  MOSES-STEINER IMAGE OF X(1113)

Barycentrics    SA (S^2-3 SB SC) ((1-J) S^2-(3+J) SB SC) : :

X(14500) lies on the Steiner deltoid and these lines: {4,13415}, {30,113}, {125,1312}, {542,8115}, {1113,2777}, {1114,5972}

X(14500) = reflection of X(i) in X(j) for these {i,j}: {125, 1312}, {1114, 5972}
X(14500) = barycentric product X(i)X(j) for these {i,j}: {1312, 11064}

X(14501)  =  MOSES-STEINER IMAGE OF X(1379)

Barycentrics    (a^2 b^2-b^4+a^2 c^2-c^4) (a^2 b^2-b^4+a^2 c^2-c^4+(b^2+c^2) Sqrt(a^4-a^2 b^2+b^4-a^2 c^2-b^2 c^2+c^4)) : :

X(14501) lies on the Steiner deltoid and these lines: {114,325}, {115,2029}, {542,6040}, {620,1380}, {626,13325}, {1340,4045}, {1379,2794}, {3558,7764}

X(14501) = reflection of X(i) in X(j) for these {i,j}: {115, 2040}, {1380, 620}
X(14501) = barycentric quotient X(2029)/X(1976)

X(14502)  =  MOSES-STEINER IMAGE OF X(1380)

Barycentrics    (a^2 b^2-b^4+a^2 c^2-c^4) (a^2 b^2-b^4+a^2 c^2-c^4-(b^2+c^2) Sqrt(a^4-a^2 b^2+b^4-a^2 c^2-b^2 c^2+c^4)) : :

X(14501) lies on the Steiner deltoid and these lines: {114,325}, {115,2028}, {542,6039}, {620,1379}, {626,13326}, {1341,4045}, {1380,2794}, {3557,7764}

X(14502) = reflection of X(i) in X(j) for these {i,j}: {115, 2039}, {1379, 620}
X(14502) = barycentric quotient X(2028)/X(1976)

X(14503)  =  MOSES-STEINER IMAGE OF X(1381)

Barycentrics    (a^2 b-b^3+a^2 c-2 a b c+b^2 c+b c^2-c^3) (a^3 b-a^2 b^2-a b^3+b^4+a^3 c+a b^2 c-a^2 c^2+a b c^2-2 b^2 c^2-a c^3+c^4+2 a Sqrt(a b c (a^3-a^2 b-a b^2+b^3-a^2 c+3 a b c-b^2 c-a c^2-b c^2+c^3))) : :

X(14503) lies on the Steiner deltoid and these lines: {11,2447}, {119,517}, {1381,2829}, {1382,3035}, {2446,10956}

X(14503) = reflection of X(1382) in X(3035)
X(14503) = trilinear product X(2447)*X(6735)

X(14504)  =  MOSES-STEINER IMAGE OF X(1382)

Barycentrics    (a^2 b-b^3+a^2 c-2 a b c+b^2 c+b c^2-c^3) (a^3 b-a^2 b^2-a b^3+b^4+a^3 c+a b^2 c-a^2 c^2+a b c^2-2 b^2 c^2-a c^3+c^4-2 a Sqrt(a b c (a^3-a^2 b-a b^2+b^3-a^2 c+3 a b c-b^2 c-a c^2-b c^2+c^3))) : :

X(14504) lies on the anticomplement of the Steiner deltoid and these lines: {11,2446}, {119,517}, {1381,3035}, {1382,2829}, {2447,10956}

X(14504) = reflection of X(1381) in X(3035)
X(14504) = trilinear product X(2446)*X(6735)

X(14505)  =  MOSES-STEINER IMAGE OF X(103)

Barycentrics    (b - c)^2*(-(a^4*b^2) + 2*a^3*b^3 - 2*a*b^5 + b^6 - a^4*c^2 - 2*a^2*b^2*c^2 + 2*a*b^3*c^2 + b^4*c^2 + 2*a^3*c^3 + 2*a*b^2*c^3 - 4*b^3*c^3 + b^2*c^4 - 2*a*c^5 + c^6) : :
X(14505) = 3 X(1565) - 2 X(14116)

X(14505) lies on the anticomplement of the Steiner deltoid and these lines: {116, 514}, {150, 927}, {1111, 3323}, {1517, 2808}

X(14505) = midpoint of X(150) and X(927)
X(14505) = reflection of X(1566) in X(116)
X(14505) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3323, 5532, 1111)
X(14505) = X(i)-isoconjugate of X(j) for these (i,j): {1110, 2724}
X(14505) = barycentric quotient X(i)/X(j) for these {i,j}: {1086, 2724}, {2808, 1252}

X(14506)  =  MOSES-STEINER IMAGE OF X(1292)

Barycentrics    (a*b - b^2 + a*c - c^2)^2*(2*a^5 - 4*a^4*b + 5*a^3*b^2 - 3*a^2*b^3 + a*b^4 - b^5 - 4*a^4*c + a^2*b^2*c + 3*b^4*c + 5*a^3*c^2 + a^2*b*c^2 - 2*a*b^2*c^2 - 2*b^3*c^2 - 3*a^2*c^3 - 2*b^2*c^3 + a*c^4 + 3*b*c^4 - c^5) : :

X(14506) lies on the anticomplement of the Steiner deltoid and these lines: {120, 518}, {3323, 4712}

X(14505) = reflection of X(5519) in X(120)

X(14507)  =  MOSES-STEINER IMAGE OF X(1293)

Barycentrics    (2*a - b - c)^2*(a^3*b^2 + a^2*b^3 - a*b^4 - b^5 - 5*a^2*b^2*c + 5*b^4*c + a^3*c^2 - 5*a^2*b*c^2 + 12*a*b^2*c^2 - 6*b^3*c^2 + a^2*c^3 - 6*b^2*c^3 - a*c^4 + 5*b*c^4 - c^5) : :

X(14507) lies on the anticomplement of the Steiner deltoid and these lines: {121, 519}, {4738, 14027}

X(14507) = reflection of X(5516) in X(121)

X(14508)  =  ANTICOMPLEMENT OF X(1553)

Barycentrics    a^16 - 12*a^12*b^4 + 25*a^10*b^6 - 15*a^8*b^8 - 6*a^6*b^10 + 10*a^4*b^12 - 3*a^2*b^14 + 14*a^12*b^2*c^2 - 15*a^10*b^4*c^2 - 36*a^8*b^6*c^2 + 59*a^6*b^8*c^2 - 21*a^4*b^10*c^2 - b^14*c^2 - 12*a^12*c^4 - 15*a^10*b^2*c^4 + 87*a^8*b^4*c^4 - 51*a^6*b^6*c^4 - 33*a^4*b^8*c^4 + 18*a^2*b^10*c^4 + 6*b^12*c^4 + 25*a^10*c^6 - 36*a^8*b^2*c^6 - 51*a^6*b^4*c^6 + 88*a^4*b^6*c^6 - 15*a^2*b^8*c^6 - 15*b^10*c^6 - 15*a^8*c^8 + 59*a^6*b^2*c^8 - 33*a^4*b^4*c^8 - 15*a^2*b^6*c^8 + 20*b^8*c^8 - 6*a^6*c^10 - 21*a^4*b^2*c^10 + 18*a^2*b^4*c^10 - 15*b^6*c^10 + 10*a^4*c^12 + 6*b^4*c^12 - 3*a^2*c^14 - b^2*c^14 : :
X(14508) = 3 X(5627) - 4 X(10264)

X(14508) lies on the the cubic K695, Steiner deltoid, and these lines: {2, 1553}, {30, 74}, {146, 3258}, {477, 5663}, {2986, 7464}

X(14508) = reflection of X(i) in X(j) for these {i,j}: {146, 3258}, {476, 74}, {14480, 477}

X(14509)  =  ANTICOMPLEMENT OF X(6071)

Barycentrics    a^2*(a - b)*(a + b)*(a - c)*(a + c)*(a^2*b^6 + a^4*b^2*c^2 - 2*a^2*b^4*c^2 + 2*b^6*c^2 - 2*a^2*b^2*c^4 - 3*b^4*c^4 + a^2*c^6 + 2*b^2*c^6) : :
X(14509) = 4 X(12833) - 3 X(13170)

X(14509) lies on the anticomplement of the Steiner deltoid and these lines: {2, 6071}, {99, 512}, {110, 3288}, {148, 2679}, {384, 8870}, {574, 6787}, {2698, 2782}, {2854, 12157}

X(14509) = reflection of X(i) in X(j) for these {i,j}: {148, 2679}, {805, 99}

X(14510)  =  ANTICOMPLEMENT OF X(6072)

Barycentrics    a^2*(a^10*b^6 - 3*a^8*b^8 + 3*a^6*b^10 - a^4*b^12 + a^12*b^2*c^2 - a^10*b^4*c^2 - 2*a^6*b^8*c^2 + 3*a^4*b^10*c^2 - a^2*b^12*c^2 - a^10*b^2*c^4 - 4*a^8*b^4*c^4 + 9*a^6*b^6*c^4 - 11*a^4*b^8*c^4 + 5*a^2*b^10*c^4 - 2*b^12*c^4 + a^10*c^6 + 9*a^6*b^4*c^6 + 3*a^4*b^6*c^6 - 2*a^2*b^8*c^6 + 5*b^10*c^6 - 3*a^8*c^8 - 2*a^6*b^2*c^8 - 11*a^4*b^4*c^8 - 2*a^2*b^6*c^8 - 6*b^8*c^8 + 3*a^6*c^10 + 3*a^4*b^2*c^10 + 5*a^2*b^4*c^10 + 5*b^6*c^10 - a^4*c^12 - a^2*b^2*c^12 - 2*b^4*c^12) : :

X(14510) lies on the anticomplement of the Steiner deltoid and these lines: {2, 6072}, {98, 385}, {147, 2679}, {576, 6785}, {2698, 2782}, {3289, 5111}

X(14510) = reflection of X(i) in X(j) for these {i,j}: {147, 2679}, {805, 98}

X(14511)  =  ANTICOMPLEMENT OF X(6073)

Barycentrics    a*(a^9 - 2*a^8*b - a^7*b^2 + 6*a^6*b^3 - 3*a^5*b^4 - 6*a^4*b^5 + 5*a^3*b^6 + 2*a^2*b^7 - 2*a*b^8 - 2*a^8*c + 12*a^7*b*c - 16*a^6*b^2*c - 12*a^5*b^3*c + 36*a^4*b^4*c - 12*a^3*b^5*c - 16*a^2*b^6*c + 12*a*b^7*c - 2*b^8*c - a^7*c^2 - 16*a^6*b*c^2 + 55*a^5*b^2*c^2 - 38*a^4*b^3*c^2 - 35*a^3*b^4*c^2 + 52*a^2*b^5*c^2 - 19*a*b^6*c^2 + 2*b^7*c^2 + 6*a^6*c^3 - 12*a^5*b*c^3 - 38*a^4*b^2*c^3 + 88*a^3*b^3*c^3 - 38*a^2*b^4*c^3 - 12*a*b^5*c^3 + 6*b^6*c^3 - 3*a^5*c^4 + 36*a^4*b*c^4 - 35*a^3*b^2*c^4 - 38*a^2*b^3*c^4 + 42*a*b^4*c^4 - 6*b^5*c^4 - 6*a^4*c^5 - 12*a^3*b*c^5 + 52*a^2*b^2*c^5 - 12*a*b^3*c^5 - 6*b^4*c^5 + 5*a^3*c^6 - 16*a^2*b*c^6 - 19*a*b^2*c^6 + 6*b^3*c^6 + 2*a^2*c^7 + 12*a*b*c^7 + 2*b^2*c^7 - 2*a*c^8 - 2*b*c^8) : :

X(14511) lies on the anticomplement of the Steiner deltoid and these lines: {1, 59}, {2, 6073}, {104, 517}, {153, 3259}, {952, 953}

X(14511) = reflection of X(i) in X(j) for these {i,j}: {153, 3259}, {901, 104}

X(14512)  =  ANTICOMPLEMENT OF X(6074)

Barycentrics    a^12 - 2*a^11*b + 3*a^10*b^2 - 10*a^9*b^3 + 12*a^8*b^4 + 2*a^7*b^5 - 8*a^6*b^6 + 2*a^5*b^7 - 5*a^4*b^8 + 8*a^3*b^9 - 3*a^2*b^10 - 2*a^11*c + 4*a^10*b*c - 4*a^7*b^4*c - 8*a^6*b^5*c + 16*a^5*b^6*c - 10*a^3*b^8*c + 4*a^2*b^9*c + 3*a^10*c^2 + a^8*b^2*c^2 - 6*a^7*b^3*c^2 - 10*a^6*b^4*c^2 + 4*a^5*b^5*c^2 + 12*a^4*b^6*c^2 + 2*a^3*b^7*c^2 - 5*a^2*b^8*c^2 - b^10*c^2 - 10*a^9*c^3 - 6*a^7*b^2*c^3 + 56*a^6*b^3*c^3 - 22*a^5*b^4*c^3 - 12*a^4*b^5*c^3 - 26*a^3*b^6*c^3 + 16*a^2*b^7*c^3 + 4*b^9*c^3 + 12*a^8*c^4 - 4*a^7*b*c^4 - 10*a^6*b^2*c^4 - 22*a^5*b^3*c^4 + 10*a^4*b^4*c^4 + 26*a^3*b^5*c^4 - 8*a^2*b^6*c^4 - 4*b^8*c^4 + 2*a^7*c^5 - 8*a^6*b*c^5 + 4*a^5*b^2*c^5 - 12*a^4*b^3*c^5 + 26*a^3*b^4*c^5 - 8*a^2*b^5*c^5 - 4*b^7*c^5 - 8*a^6*c^6 + 16*a^5*b*c^6 + 12*a^4*b^2*c^6 - 26*a^3*b^3*c^6 - 8*a^2*b^4*c^6 + 10*b^6*c^6 + 2*a^5*c^7 + 2*a^3*b^2*c^7 + 16*a^2*b^3*c^7 - 4*b^5*c^7 - 5*a^4*c^8 - 10*a^3*b*c^8 - 5*a^2*b^2*c^8 - 4*b^4*c^8 + 8*a^3*c^9 + 4*a^2*b*c^9 + 4*b^3*c^9 - 3*a^2*c^10 - b^2*c^10 : :

X(14512) lies on the anticomplement of the Steiner deltoid and these lines: {2, 6074}, {103, 516}, {152, 1566}, {2724, 2808}

X(14512) = reflection of X(i) in X(j) for these {i,j}: {152, 1566}, {927, 103}

X(14513)  =  ANTICOMPLEMENT OF X(6075)

Barycentrics    a*(a - b)*(a - c)*(a^3 - a^2*b + 2*b^3 - a^2*c + a*b*c - 2*b^2*c - 2*b*c^2 + 2*c^3) : :

X(14513) lies on the anticomplement of the Steiner deltoid and these lines: {1, 1168}, {2, 6075}, {36, 899}, {59, 110}, {100, 513}, {101, 4893}, {149, 3259}, {404, 10428}, {484, 5524}, {517, 3935}, {661, 1252}, {952, 953}, {1290, 8701}, {1319, 7292}, {3006, 5176}, {3240, 5091}, {3952, 4076}, {4998, 7192}, {5380, 5385}, {6550, 6634}

X(14513) = reflection of X(i) in X(j) for these {i,j}: {149, 3259}, {901, 100}
X(14513) = {X(2222),X(4551)}-harmonic conjugate of X(59)

X(14514)  =  ANTICOMPLEMENT OF X(6076)

Barycentrics    (a - b)*(a + b)*(a - c)*(a + c)*(a^12 - 7*a^10*b^2 + 9*a^8*b^4 + 42*a^6*b^6 + 23*a^4*b^8 - 3*a^2*b^10 - b^12 - 7*a^10*c^2 + 55*a^8*b^2*c^2 - 128*a^6*b^4*c^2 - 415*a^4*b^6*c^2 + 97*a^2*b^8*c^2 - 2*b^10*c^2 + 9*a^8*c^4 - 128*a^6*b^2*c^4 + 960*a^4*b^4*c^4 - 126*a^2*b^6*c^4 + b^8*c^4 + 42*a^6*c^6 - 415*a^4*b^2*c^6 - 126*a^2*b^4*c^6 + 4*b^6*c^6 + 23*a^4*c^8 + 97*a^2*b^2*c^8 + b^4*c^8 - 3*a^2*c^10 - 2*b^2*c^10 - c^12) : :

X(14514) lies on the anticomplement of the Steiner deltoid and these lines: {2, 6076}, {1296, 1499}

X(14514) = reflection of X(6082) in X(1296)

X(14515)  =  ANTICOMPLEMENT OF X(6077)

Barycentrics    a^12 - 2*a^10*b^2 + 3*a^8*b^4 + 13*a^6*b^6 + 4*a^4*b^8 - 3*a^2*b^10 - 2*a^10*c^2 - 16*a^8*b^2*c^2 + 17*a^6*b^4*c^2 - 98*a^4*b^6*c^2 + 32*a^2*b^8*c^2 - b^10*c^2 + 3*a^8*c^4 + 17*a^6*b^2*c^4 + 93*a^4*b^4*c^4 + 3*a^2*b^6*c^4 - 4*b^8*c^4 + 13*a^6*c^6 - 98*a^4*b^2*c^6 + 3*a^2*b^4*c^6 - 6*b^6*c^6 + 4*a^4*c^8 + 32*a^2*b^2*c^8 - 4*b^4*c^8 - 3*a^2*c^10 - b^2*c^10 : :

X(14515) lies on the anticomplement of the Steiner deltoid and these lines: {2, 6077}, {111, 524}

X(14515) = reflection of X(6082) in X(111)

X(14516)  =  ANTICOMPLEMENT OF X(6146)

Barycentrics    2 a^10-5 a^8 b^2+4 a^6 b^4-2 a^4 b^6+2 a^2 b^8-b^10-5 a^8 c^2+6 a^6 b^2 c^2-2 a^4 b^4 c^2-2 a^2 b^6 c^2+3 b^8 c^2+4 a^6 c^4-2 a^4 b^2 c^4-2 b^6 c^4-2 a^4 c^6-2 a^2 b^2 c^6-2 b^4 c^6+2 a^2 c^8+3 b^2 c^8-c^10 : :
X(14516) = 3 X(3060) - 4 X(6756) = 2 X(52) - 3 X(7576) = 3 X(51) - 2 X(10112) = 3 X(9730) - 2 X(10116) = 3 X(7540) - 2 X(10263) = 4 X(9825) - 3 X(11245) = 3 X(5946) - 2 X(11264) = 4 X(5) - 3 X(12022) = 17 X(7486) - 12 X(12024) = 5 X(3091) - 4 X(12241) = 3 X(11459) - X(12289) = 5 X(11444) - 4 X(12362) = 3 X(381) - 2 X(12370) = 3 X(11459) - 2 X(12605) = 3 X(428) - 2 X(13142) = 5 X(3567) - 4 X(13292) = 5 X(11439) - 4 X(13488)

X(14516) lies on the cubic K364 and these lines: {2,6146}, {3,70}, {4,155}, {5,49}, {6,7544}, {20,64}, {22,9833}, {24,68}, {25,12429}, {30,11412}, {51,10112}, {52,539}, {140,11449}, {156,10024}, {184,13160}, {185,542}, {186,12359}, {343,1601}, {378,12118}, {381,12370}, {403,9927}, {428,13142}, {467,8884}, {548,11454}, {550,11440}, {578,5133}, {858,1092}, {1147,1594}, {1216,11750}, {1352,7503}, {1511,13561}, {2071,6247}, {2930,7527}, {3060,6756}, {3091,11427}, {3167,7507}, {3292,11572}, {3410,14118}, {3520,12302}, {3547,6800}, {3549,9707}, {3564,3575}, {3567,13292}, {3818,11424}, {5012,7399}, {5422,7401}, {5449,10018}, {5562,12225}, {5654,7547}, {5946,11264}, {6198,12428}, {6240,11660}, {6243,11819}, {6515,7487}, {6776,6815}, {7394,10982}, {7486,12024}, {7540,10263}, {7542,11464}, {7577,9820}, {7583,11447}, {7584,11448}, {7689,10295}, {9730,10116}, {9825,11245}, {11413,14216}, {11439,13488}, {11444,12362}, {11452,11542}, {11453,11543}, {11459,12289}, {11550,13346}, {11591,12363}, {12038,12827}, {12162,12292}, {12164,12173}, {12168,14130}

X(14516) = complement of X(34799)
X(14516) = cevapoint of X(155) and X(2917)
X(14516) = crosspoint, wrt excentral or tangential triangle, of X(155) and X(2917)
X(14516) = X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4, 6193, 1993), (5, 54, 14389), (24, 68, 3580), (49, 6288, 5), (2888, 7488, 343), (9927, 10539, 403), (11459, 12289, 12605)

X(14517)  =  ISOGONAL CONJUGATE OF X(8906)

Barycentrics    a^2 (a^2+b^2-c^2) (a^2-b^2+c^2) (a^4-2 a^2 b^2+b^4-2 a^2 c^2+c^4) (a^12-2 a^10 b^2-a^8 b^4+4 a^6 b^6-a^4 b^8-2 a^2 b^10+b^12-6 a^10 c^2+10 a^8 b^2 c^2-8 a^6 b^4 c^2+8 a^4 b^6 c^2-2 a^2 b^8 c^2-2 b^10 c^2+15 a^8 c^4-8 a^6 b^2 c^4+2 a^4 b^4 c^4+8 a^2 b^6 c^4-b^8 c^4-20 a^6 c^6-8 a^4 b^2 c^6-8 a^2 b^4 c^6+4 b^6 c^6+15 a^4 c^8+10 a^2 b^2 c^8-b^4 c^8-6 a^2 c^10-2 b^2 c^10+c^12) (a^12-6 a^10 b^2+15 a^8 b^4-20 a^6 b^6+15 a^4 b^8-6 a^2 b^10+b^12-2 a^10 c^2+10 a^8 b^2 c^2-8 a^6 b^4 c^2-8 a^4 b^6 c^2+10 a^2 b^8 c^2-2 b^10 c^2-a^8 c^4-8 a^6 b^2 c^4+2 a^4 b^4 c^4-8 a^2 b^6 c^4-b^8 c^4+4 a^6 c^6+8 a^4 b^2 c^6+8 a^2 b^4 c^6+4 b^6 c^6-a^4 c^8-2 a^2 b^2 c^8-b^4 c^8-2 a^2 c^10-2 b^2 c^10+c^12) : :

X(14517) lies on the cubic K919 and this line: {24,6193}

X(14517) = isogonal conjugate of X(8906)
X(14517) = X(3)-cross conjugate of X(24)
X(14517) = X(4)-vertex conjugate of X(6193)
X(14517) = X(i)-isoconjugate of X(j) for these (i,j): {1, 8906}, {91, 9937}
X(14517) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 8906}, {571, 9937}

X(14518)  =  ISOGONAL CONJUGATE OF X(8905)

Barycentrics    a^2 (a^2+b^2-c^2)^2 (a^2-b^2+c^2)^2 (a^4-2 a^2 b^2+b^4-a^2 c^2-b^2 c^2) (a^4-a^2 b^2-2 a^2 c^2-b^2 c^2+c^4) (a^8-2 a^6 b^2+4 a^4 b^4-6 a^2 b^6+3 b^8-4 a^6 c^2+2 a^4 b^2 c^2-6 b^6 c^2+6 a^4 c^4+2 a^2 b^2 c^4+4 b^4 c^4-4 a^2 c^6-2 b^2 c^6+c^8) (a^8-4 a^6 b^2+6 a^4 b^4-4 a^2 b^6+b^8-2 a^6 c^2+2 a^4 b^2 c^2+2 a^2 b^4 c^2-2 b^6 c^2+4 a^4 c^4+4 b^4 c^4-6 a^2 c^6-6 b^2 c^6+3 c^8) : :

X(14518) lies on the cubic K919 and these lines: {24,8883}, {317,6193}

X(14518) = isogonal conjugate of X(8905)
X(14518) = X(3)-cross conjugate of X(8884)
X(14518) = X(4)-vertex conjugate of X(8883).
X(14518) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 8905}, {8882, 6193}

X(14519)  =  X(165)X(1202)∩X(354)X(10481)

Trilinears    ((b+c)*a-(b-c)^2)*(2*(b+c)*a^ 3-(4*b^2-b*c+4*c^2)*a^2+2*(b^2-c^2)*(b-c)*a-b*c*(b-c)^2) : :

See César Lozada, Hyacinthos 26620

X(14519) lies on these lines: {165, 1202}, {354, 10481}, {991, 995}, {1362, 5919}, {5049, 5542}

X(14520)  =  X(1)X(1362)∩X(3)X(6)

Trilinears    a*((b+c)^2*a^4-2*(b^3+c^3)*a^ 3-2*b*c*(2*b^2-b*c+2*c^2)*a^2+ 2*(b^3-c^3)*(b^2-c^2)*a-(b^4+ c^4)*(b-c)^2) : :
X(14520) = (4*R*s^2-SW*(4*R+r))*X(3)+SW*( 4*R+r)*X(6)

See César Lozada, Hyacinthos 26620

X(14520) lies on these lines: {1, 1362}, {3, 6}, {51, 11349}, {942, 10481}, {946, 971}, {3333, 4334}

X(14520) = = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (991, 4253, 3), (6479, 8410, 1030)

X(14521)  =  X(57)X(71)∩X(65)X(5223)

Trilinears    (a^5-3*(b+c)*a^4+2*(b^2+4*b*c+ c^2)*a^3+2*(b+c)*(b^2-10*b*c+c^2)*a^2-(b+c)*(b^2+6*b*c+c^2) *(3*(b+c)*a-(b-c)^2))*(a-b+c)* (a+b-c) : :

See César Lozada, Hyacinthos 26620

X(14521) lies on these lines: {57, 71}, {65, 5223}

X(14522)  =  X(1)X(7955)∩X(33)X(8917)

Trilinears    (-a+b+c)*(a^6-3*(b-c)^2*a^4-16*(b^2-c^2)*(b-c)*b*c*a+(b^2+6*b*c+c^2)*(b-c)^2*(3*a^2-(b-c)^2)) : :

See César Lozada, Hyacinthos 26620

X(14522) lies on these lines: {1, 7955}, {33, 8917}, {282, 522}, {354, 4328}, {3938, 4319}, {10703, 11224}

X(14523)  =  X(1)X(6)∩X(65)X(13572)

Trilinears    (b+c)*a^3-(3*b^2-2*b*c+3*c^2)*a^2+(b+c)*(3*b^2-4*b*c+3*c^2)*a-(b^2+c^2)*(b-c)^2 : :
X(14523) = 3*X(354)-X(1122)

See César Lozada, Hyacinthos 26620

X(14523) lies on these lines: {1, 6}, {65, 13572}, {354, 1122}, {938, 5015}, {942, 4307}, {971, 4310}, {982, 1742}, {2835, 12016}, {2887, 11019}, {3008, 3059}, {3663, 14100}, {3672, 7671}, {3677, 10391}, {3945, 11025}, {3976, 4334}, {4388, 10580}, {9309, 12915}, {9844, 13161}

X(14524)  =  X(6)X(57)∩X(65)X(516)

Trilinears    (a+b-c)*(a-b+c)*((b+c)*a^4-4*b*c*a^3-2*(b+c)*(b^2-3*b*c+c^2)*a^2+(b^4-c^4)*(b-c)) : :

See César Lozada, Hyacinthos 26620

X(14524) lies on these lines: {6, 57}, {65, 516}, {77, 4254}, {169, 6180}, {198, 4341}, {241, 573}, {1020, 1108}, {1214, 2269}, {1400, 9502}, {1828, 1876}, {2262, 3668}, {5120, 7013}, {8270, 10387}

X(14525)  =  X(527)X(4428)∩X(910)X(2285)

Trilinears    (a^6-2*(b+c)*a^5+(b^2+4*b*c+c^2)*a^4-2*(b+c)*b*c*a^3-(b^4+c^4-2*(b^2-13*b*c+c^2)*b*c)*a^2+ 2*(b^3+c^3)*(b-c)^2*a-(b^4-10*b^2*c^2+c^4)*(b-c)^2)*a : :

See César Lozada, Hyacinthos 26620

X(14525) lies on these lines: {527, 4428}, {910, 2285}, {1279, 3304}, {1486, 4644}, {3340, 3827}

X(14526)  = CENTER OF THE 1st SCHIFFLER CIRCLE

Trilinears    2*(cos(A)+1)+(cos(B)+cos(C))*(4*sin(B)*sin(C)+1)+cos(B-C) : :
Barycentrics    (b+c)*a^6+2*b*c*a^5-(b+c)*(3*b^2-b*c+3*c^2)*a^4-b*c*(3*b^2+4*b*c+3*c^2)*a^3+3*(b^3-c^3)*(b^2-c^2)*a^2+(b^2-c^2)^2*b*c*a-(b^2-c^2)^3*(b-c) : :

X(14526) = X(5086)-3*X(6175)

Let A'B'C' be the 1st Schiffler triangle of ABC. Let Aa', Bb', Cc' be the orthogonal projections of A, B, C on B'C', C'A', A'B', respectively, and A'a, B'b, C'c the orthogonal projections of A', B', C' on BC, CA, AB. These six points lie on a circle here named the 1st Schiffler circle, with center X(14526) and squared radius ((9*R+2*r)*R^2*r^2+(R+2*r)*S^2)*R/((R+2*r)^2*(3*R+2*r)^2). This circle passes through X(11) and X(1365) and is the circumcircle of the pedal triangles of the isogonal conjugate pair X(35) and X(79).

Next, if "1st Schiffler triangle" is replaced by "2nd Schiffler triangle", the resulting six points lie on a circle, here named the 2nd Schiffler circle, with center X(1737) and squared radius R*r^2/(R-2*r). This circle passes through X(11), X(5532), X(13141), X(14027) and is the circumcircle of the pedal triangles of the isogonal conjugate pair X(36) and X(80).

Continuing, let A'B'C' be the 1st Schiffler triangle and A"B"C" the 2nd Schiffler triangle of ABC. Let A1, B1, C1 be the orthogonal projections of A', B', C' on B"C", C"A", A"B", respectively, and A2, B2, C2 the orthogonal projections of A", B", C" on B'C', C'A' and A'B'. These six points A1, B1, C1 , A2, B2, C2 lie on a circle here named the 3rd Schiffler circle, with center X(14527). This circle also passes through X(11).

Contributed by César Lozada, September 23, 2017)

For definitions of Schiffler triangles, see See César Lozada, Hyacinthos 26620

X(14526) lies on these lines: {1,149}, {11,13852}, {12,2771}, {30,2646}, {35,79}, {65,5499}, {191,329}, {442,1737}, {517,3649}, {758,10039}, {908,3647}, {946,5441}, {1155,11277}, {3065,6888}, {4295,10056}, {5086,6175}, {5219,7701}, {5249,6701}, {5902,6937}, {6825,10044}, {8256,10954}, {11375,13743}, {12866,13089}

X(14526) = midpoint of X(35) and X(79)
X(14526) = {X(79), X(3651)}-harmonic conjugate of X(1770)

X(14527) = CENTER OF THE 3rd SCHIFFLER CIRCLE

Barycentrics    (b^2+c^2)*a^8-2*(b^3+c^3)*a^7-2*(b^2+c^2)*(b^2-b*c+c^2)*a^6+2*(b^2-c^2)*(b-c)*(3*b^2+b*c+3*c^2)*a^5-b*c*(-3*b*c^3+2*c^4+2*b^4-3*b^3*c)*a^4-2*(b+c)*(3*c^6-7*b*c^5+5*b^2*c^4-3*b^3*c^3+5*b^4*c^2-7*b^5*c+3*b^6)*a^3+(b^2-c^2)^2*a^2*(2*b^4-2*b^3*c+b^2*c^2-2*b*c^3+2*c^4)+2*(b^2-c^2)^3*(b-c)*a*(b^2-b*c+c^2)-(b^2-c^2)^4*(b-c)^2 : :

See X(14526).

X(14527) lies on these lines: {5,758}, {21,3816}, {442,5443}, {6841,12600}, {10021,12623}

X(14528) = ISOGONAL CONJUGATE OF X(3091)

Barycentrics    (S^2+2*SA*SB)*(S^2+2*SA*SC)*(SB+SC) : :

Let A'B'C' be the midheight triangle. X(145287) is the radical center of the circumcircles of A'BC, B'CA, C'AB. (Randy Hutson, December 2, 2017)

X(14528) lies on the Jerabek hyperbola, the cubic K918, and these lines: {3,13382}, {4,154}, {6,3515}, {24,1173}, {54,9786}, {64,184}, {68,140}, {69,3523}, {72,3576}, {73,5204}, {74,1181}, {182,6391}, {185,3532}, {186,13472}, {248,5013}, {265,1656}, {371,6416}, {372,6415}, {468,14457}, {550,4846}, {578,3517}, {1151,6414}, {1152,6413}, {1176,1350}, {1192,11402}, {1498,3426}, {1657,3521}, {1903,2261}, {3431,7592}, {3520,13452}, {3522,3796}, {3526,7666}, {3531,10282}, {5059,6800}, {5094,6145}, {5609,7526}, {7488,11477}, {9704,11559}, {10601,11449}, {10605,11270}, {10982,11464}, {11202,11426}, {11456,11738}

X(14528) = isogonal conjugate of X(3091)

X(14529) =  MIDPOINT OF X(221) AND X(3556)

Barycentrics    a^2*(a^5 + a^4*b - a^3*b^2 - a^2*b^3 + a^4*c - b^4*c - a^3*c^2 + b^3*c^2 - a^2*c^3 + b^2*c^3 - b*c^4) : :
X(14529) = 3 X(154) + X(221) = 3 X(154) - X(3556)

X(14529) lies on the cubic K838 and these lines: {1, 1437}, {6, 2333}, {31, 56}, {40, 692}, {47, 859}, {65, 184}, {110, 3869}, {182, 3812}, {197, 7078}, {205, 4559}, {206, 942}, {341, 4579}, {517, 1147}, {578, 7686}, {595, 3941}, {958, 3955}, {960, 9306}, {994, 9563}, {1092, 14110}, {1104, 1397}, {1385, 5248}, {1426, 1456}, {1455, 7335}, {1498, 8053}, {1626, 4303}, {1854, 10535}, {1858, 11363}, {2360, 3185}, {2361, 13738}, {2778, 11699}, {2818, 10282}, {3157, 8679}, {4999, 10192}, {5301, 7113}, {5887, 10539}, {6000, 6097}, {6987, 12930}, {7299, 13724}

X(14529) = midpoint of X(i) and X(j) for these {i,j}: {221, 3556}, {3157, 9798}
X(14529) = X(1790)-Ceva conjugate of X(6)
X(14529) = crosssum of X(523) and X(2968)
X(14529) = X(21)-beth conjugate of X(2217)
X(14529) = {X(154),X(221)}-harmonic conjugate of X(3556)

X(14530) =  MIDPOINT OF X(1498) AND X(8567)

Barycentrics    a^2*(5*a^8 - 14*a^6*b^2 + 12*a^4*b^4 - 2*a^2*b^6 - b^8 - 14*a^6*c^2 + 4*a^4*b^2*c^2 + 2*a^2*b^4*c^2 + 8*b^6*c^2 + 12*a^4*c^4 + 2*a^2*b^2*c^4 - 14*b^4*c^4 - 2*a^2*c^6 + 8*b^2*c^6 - c^8) : :
X(14530) = 7 X(3) - 2 X(64) = X(3) - 6 X(154) = 4 X(159) + X(1351) = 9 X(154) + X(1498) = 3 X(3) + 2 X(1498) = 3 X(64) + 7 X(1498) = X(1657) + 4 X(2883) = 8 X(156) - 3 X(3167) = 9 X(64) - 14 X(3357) = 9 X(3) - 4 X(3357) = 3 X(1498) + 2 X(3357) = 8 X(206) - 3 X(5050) = 6 X(1853) - 11 X(5070) = 2 X(550) + 3 X(5656) = 3 X(3534) + 2 X(5878) = 3 X(382) - 8 X(5893) = 4 X(548) + X(6225) = 9 X(5054) - 4 X(6247) = X(1498) - 6 X(6759) = 3 X(154) + 2 X (6759) = X(3) + 4 X(6759) = X(3357) + 9 X(6759) = X(64) + 14 X(6759) = 4 X(156) + X(7387) = 3 X(3167) + 2 X(7387) = 9 X(5050) - 4 X(8549) = 6 X(206) - X(8549) = 3 X(64) - 7 X(8567) = 2 X(3357) - 3 X(8567) = 3 X(3) - 2 X(8567) = 9 X(154) - X(8567) = 6 X(6759) + X(8567) = 3 X(381) + 2 X(9833) = 2 X(155) + 3 X(9909) = 4 X(110) + X(9919) = 2 X(195) + 3 X(9920) = 3 X(5093) + 2 X(9924) = 9 X(9920) - 4 X(9935) = 3 X(195) + 2 X(9935) = 7 X(3526) - 12 X(10192) = 3 X(3) - 8 X(10282) = X(3357) - 6 X(10282) = 9 X(154) - 4 X(10282) = X(8567) - 4 X(10282) = 3 X(6759) + 2 X(10282) = X(1498) + 4 X(10282) = 11 X(8567) - 9 X(10606) = 11 X(3) - 6 X(10606) = 11 X(154) - X(10606) = 11 X(1498) + 9 X(10606) = 4 X(10537) + X(10679) = 7 X(8567) - 18 X(11202) = 7 X(3) - 12 X(11202) = 14 X(10282) - 9 X(11202) = X(64) - 6 X(11202) = 7 X(154) - 2 X(11202) = 7 X(6759) + 3 X(11202) = 7 X(1498) + 18 X(11202) = 17 X(8567) - 18 X(11204) = 17 X(3) - 12 X(11204) = 17 X(11202) - 7 X(11204) = 17 X(154) - 2 X(11204) = 17 X(6759) + 3 X(11204) = 17 X(1498) + 18 X(11204) = 2 X(5) + 3 X(11206) = 6 X(10154) - X(11411) = 4 X(26) + X(12164) = 6 X(8703) - X(12250) = 8 X(1498) - 3 X(12315) = 16 X(6759) - X(12315) = 4 X(3) + X(12315) = 8 X(8567) + 3 X(12315) = 8 X(64) + 7 X(12315) = 16 X(3357) + 9 X(12315) = 6 X(549) - X(12324) = 12 X(64) - 7 X(13093) = 8 X(3357) - 3 X(13093) = 6 X(3) - X(13093) = 4 X(8567) - X(13093) = 16 X(10282) - X(13093) = 4 X(1498) + X(13093) = 3 X(12315) + 2 X(13093) = X(12308) + 4 X(13289) = 7 X(3526) - 2 X(14216) = 6 X(10192) - X(14216)

X(14530) lies on these lines: {3, 64}, {5, 11206}, {23, 12160}, {25, 1614}, {26, 12164}, {49, 1660}, {54, 5198}, {110, 9919}, {155, 9909}, {156, 3167}, {159, 195}, {182, 11484}, {184, 1598}, {186, 12174}, {206, 5050}, {381, 9833}, {382, 5893}, {548, 6225}, {549, 12324}, {550, 5656}, {1181, 1495}, {1503, 1656}, {1593, 9707}, {1657, 2883}, {1853, 5070}, {2393, 11482}, {3295, 10535}, {3311, 10533}, {3312, 10534}, {3426, 3516}, {3515, 11456}, {3526, 10192}, {3527, 10594}, {3534, 5878}, {5054, 6247}, {5093, 9924}, {6090, 10323}, {6221, 12970}, {6398, 12964}, {6427, 11241}, {6428, 11242}, {6800, 7395}, {7691, 9715}, {8703, 12250}, {9652, 10833}, {9818, 10610}, {10154, 11411}, {10306, 10536}, {10537, 10679}, {11410, 12290}, {11413, 11820}, {12308, 13289}

X(14530) = midpoint of X(1498) and X(8567)
X(14530) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 1498, 13093), (64, 11202, 3), (154, 1498, 10282), (154, 6759, 3), (156, 7387, 3167), (159, 7517, 9920), (184, 1598, 11426), (1181, 1495, 3517), (1498, 10282, 3), (1498, 13093, 12315), (6759, 10282, 1498), (9707, 14157, 1593), (10192, 14216, 3526), (10594, 11402, 3527)

X(14531) =  REFLECTION OF X(185) IN X(5889)

Barycentrics    a^2*(a^2*b^2 - b^4 + a^2*c^2 + 2*b^2*c^2 - c^4)*(3*a^4 - 6*a^2*b^2 + 3*b^4 - 6*a^2*c^2 + 2*b^2*c^2 + 3*c^4) : :
X(14531) = 8 X(5) - 9 X(51) = 3 X(51) - 4 X(52) = 2 X(5) - 3 X(52) = 15 X(51) - 16 X(143) = 5 X(5) - 6 X(143) = 5 X(52) - 4 X(143) = 2 X(20) - 3 X(185) = 6 X(389) - 5 X(631) = 3 X(568) - 2 X(1216) = 9 X(568) - 7 X(3526) = 6 X(1216) - 7 X(3526) = 9 X(373) - 10 X(3567) = 9 X(3060) - 7 X(3832) = 10 X(631) - 9 X(3917) = 4 X(389) - 3 X(3917) = 15 X(3567) - 13 X(5067) = 5 X(3843) - 6 X(5446) = 11 X(5070) - 12 X(5462) = 8 X(143) - 5 X(5562) = 4 X(5) - 3 X(5562) = 3 X(51) - 2 X(5562) = 8 X(1216) - 9 X(5650) = 4 X(568) - 3 X(5650) = 4 X(3861) - 3 X(5876) = X(20) - 3 X(5889) = 7 X(3528) - 9 X(5890) = 10 X(5) - 9 X(5891) = 5 X(5562) - 6 X(5891) = 5 X(51) - 4 X(5891) = 5 X(52) - 3 X(5891) = 4 X(143) - 3 X(5891) = 7 X(3832) - 6 X(5907) = 3 X(3060) - 2 X(5907) = 4 X(3530) - 3 X(6101) = 2 X(548) - 3 X(6102) = X(382) - 3 X(6243) = 18 X(5943) - 17 X(7486) = 3 X(2979) - 4 X(9729) = 8 X(3530) - 9 X(9730) = 2 X(6101) - 3 X(9730) = 11 X(5562) - 16 X(10095) = 11 X(5) - 12 X(10095) = 11 X(143) - 10 X(10095) = 11 X(52) - 8 X(10095) = 11 X(3855) - 12 X(10110) = 2 X(3853) - 3 X(10263) = 4 X(548) - 3 X(10625) = 4 X(382) - 3 X(11381) = 4 X(6243) - X(11381) = 5 X(631) - 3 X(11412) = 3 X(3917) - 2 X(11412) = 17 X(7486) - 15 X(11444) = 6 X(5943) - 5 X(11444) = 11 X(3855) - 9 X(11459) = 4 X(10110) - 3 X(11459) = 14 X(10095) - 11 X(11591) = 7 X(5562) - 8 X(11591) = 7 X(5) - 6 X(11591) = 7 X(143) - 5 X(11591) = 7 X(52) - 4 X(11591) = 7 X(7999) - 8 X(11695) = 13 X(5067) - 12 X(11793) = 9 X(373) - 8 X(11793) = 5 X(3567) - 4 X(11793) = 4 X(3853) - 3 X(12162) = 5 X(10574) - 4 X(13348) = 17 X(5) - 18 X(13364) = 17 X(51) - 16 X(13364) = 17 X(143) - 15 X(13364) = 17 X(52) - 12 X(13364) = 3 X(376) - 4 X(13382) = 5 X(5907) - 6 X(13570) = 5 X(3060) - 4 X(13570) = 13 X(5562) - 16 X(14128) = 13 X(11591) - 14 X(14128) = 13 X(5) - 12 X(14128) = 13 X(10095) - 11 X(14128) = 13 X(143) - 10 X(14128) = 13 X(52) - 8 X(14128) = 2 X(3861) - 3 X(14449)

X(14531) lies on these lines: {3, 13366}, {5, 51}, {20, 185}, {24, 3292}, {49, 12316}, {68, 11572}, {155, 1495}, {184, 9715}, {373, 3567}, {376, 13382}, {382, 6243}, {389, 631}, {524, 3575}, {548, 6102}, {567, 12307}, {568, 1216}, {576, 7503}, {1351, 11424}, {1593, 8541}, {1907, 3867}, {1993, 13367}, {1994, 7691}, {2393, 6293}, {2807, 9589}, {2979, 9729}, {3060, 3832}, {3270, 9643}, {3528, 5890}, {3530, 6101}, {3548, 13857}, {3581, 12038}, {3627, 13421}, {3843, 5446}, {3853, 10263}, {3855, 10110}, {3861, 5876}, {5070, 5462}, {5097, 13434}, {5943, 7486}, {5965, 14516}, {6238, 9644}, {7488, 9706}, {7999, 11695}, {8799, 11197}, {9705, 10282}, {10112, 12225}, {10574, 13348}, {10575, 13391}, {11411, 11550}, {11800, 12219}, {12111, 13598}

X(14531) = reflection of X(i) in X(j) for these {i,j}: {185, 5889}, {3627, 13421}, {5562, 52}, {5876, 14449}, {10625, 6102}, {11412, 389}, {12111, 13598}, {12162, 10263}, {12219, 11800}, {12225, 10112}
X(14531) = X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (52, 5562, 51), (52, 5891, 143), (389, 11412, 3917), (3567, 11793, 373)
X(14531) = X(13157)-Ceva conjugate of X(216)
X(14531) = crossdifference of every pair of points on line {2451, 2623}
X(14531) = barycentric product X(343)*X(3515)
X(14531) = barycentric quotient X(3515)/X(275)

X(14532) =  REFLECTION OF X(11159) IN X(376)

Barycentrics    a^8 + 9*a^6*b^2 - 5*a^4*b^4 - 5*a^2*b^6 + 9*a^6*c^2 - 2*a^4*b^2*c^2 - 3*a^2*b^4*c^2 - 4*b^6*c^2 - 5*a^4*c^4 - 3*a^2*b^2*c^4 + 8*b^4*c^4 - 5*a^2*c^6 - 4*b^2*c^6 : :
X(14532) = 4 X(3) - 3 X(11286) = 2 X(4) - 3 X(11287) = 4 X(10796) - 5 X(12017) = 5 X(3522) - 3 X(14033)

X(14532) lies on the cubic K820 and these lines: {2, 3}, {7776, 8721}, {8722, 9756}, {10796, 12017}

X(14532) = reflection of X(i) in X(j) for these {i,j}: {11159, 376}, {11356, 7470}
X(14532) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (376, 13860, 3), (3522, 11285, 3)

X(14533) =  ISOGONAL CONJUGATE OF X(324)

Barycentrics    a^4*(a^2 - b^2 - c^2)*(a^4 - 2*a^2*b^2 + b^4 - a^2*c^2 - b^2*c^2)*(a^4 - a^2*b^2 - 2*a^2*c^2 - b^2*c^2 + c^4) : :

X(14533) lies on the cubic K378 and these lines: {6, 24}, {53, 1970}, {95, 141}, {96, 6146}, {97, 394}, {160, 184}, {216, 5961}, {217, 2965}, {231, 10619}, {233, 6689}, {275, 1971}, {570, 13367}, {577, 1147}, {1181, 8883}, {1409, 2148}, {2169, 3990}, {2963, 8579}, {5421, 8779}, {8745, 9707}, {8901, 9722}

X(14533) = isogonal conjugate of X(324)
X(14533) = X(184)-cross conjugate of X(54)
X(14533) = X(i)-isoconjugate of X(j) for these (i,j): {1, 324}, {4, 14213}, {5, 92}, {19, 311}, {51, 1969}, {53, 75}, {63, 13450}, {76, 2181}, {91, 467}, {158, 343}, {264, 1953}, {275, 1087}, {331, 7069}, {561, 3199}, {648, 2618}, {811, 12077}, {823, 6368}, {1393, 7017}, {1895, 13157}, {2962, 14129}, {5562, 6521}
X(14533) = crosspoint of X(i) and X(j) for these (i,j): {54, 97}, {96, 275}
X(14533) = crosssum of X(i) and X(j) for these (i,j): {5, 53}, {52, 216}, {571, 10274}
X(14533) = {X(54),X(8882)}-harmonic conjugate of X(6)
X(14533) = barycentric product X(i)*X(j) for these {i,j}: {1, 2169}, {3, 54}, {6, 97}, {48, 2167}, {49, 252}, {63, 2148}, {95, 184}, {96, 1147}, {255, 2190}, {275, 577}, {323, 11077}, {394, 8882}, {520, 933}, {1092, 8884}, {1166, 1216}, {2616, 4575}, {2623, 4558}, {6759, 14371}
X(14533) = barycentric quotient X(i)/X(j) for these {i,j}: {3, 311}, {6, 324}, {25, 13450}, {32, 53}, {48, 14213}, {54, 264}, {97, 76}, {184, 5}, {560, 2181}, {571, 467}, {577, 343}, {810, 2618}, {933, 6528}, {1216, 1225}, {1501, 3199}, {2148, 92}, {2167, 1969}, {2169, 75}, {2965, 14129}, {3049, 12077}, {8882, 2052}, {9247, 1953}, {11077, 94}

X(14534) =  ISOGONAL CONJUGATE OF X(2092)

Barycentrics    (a + b)*(a + c)*(a^2 + b^2 + a*c + b*c)*(a^2 + a*b + b*c + c^2) : :

X(14534) lies on the Kiepert hyperbola, the cubic K379, and these lines: {2, 261}, {3, 3597}, {4, 1798}, {6, 7058}, {10, 58}, {21, 961}, {76, 940}, {81, 314}, {86, 226}, {99, 3666}, {286, 1396}, {572, 2051}, {741, 1961}, {1171, 6539}, {1326, 6685}, {4080, 8025}, {4581, 5466}, {5061, 5080}, {5263, 6043}

X(14534) = isogonal conjugate of X(2092)
X(14534) = isotomic conjugate of X(1211)
X(14534) = polar conjugate of X(429)
X(14534) = Cundy-Parry Phi transform of X(3597)
X(14534) = Cundy-Parry Psi transform of X(13323)
X(14534) = X(i)-cross conjugate of X(j) for these (i,j): {6, 961}, {513, 99}, {2298, 2363}, {4391, 648}, {5051, 75}, {6588, 107}, {6703, 2}
X(14534) = X(i)-isoconjugate of X(j) for these (i,j): {1, 2092}, {2, 3725}, {6, 2292}, {10, 2300}, {19,22076}, {31, 1211}, {37, 1193}, {42, 3666}, {48, 429}, {65, 2269}, {71, 1829}, {72, 2354}, {213, 4357}, {228, 1848}, {512, 3882}, {560, 1228}, {604, 3704}, {960, 1400}, {1018, 6371}, {1042, 3965}, {1169, 6042}, {1402, 3687}, {2171, 4267}
X(14534) = cevapoint of X(i) and X(j) for these (i,j): {2, 81}, {6, 21}, {58, 572}, {1169, 1798}, {1220, 2298}
X(14534) = trilinear pole of line {523, 1325}
X(14534) = barycentric product X(i)*X(j) for these {i,j}: {58, 1240}, {75, 2363}, {76, 1169}, {86, 1220}, {99, 4581}, {264, 1798}, {274, 2298}, {286, 1791}, {314, 961}, {4560, 6648}, {7192, 8707}
X(14534) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 2292}, {2, 1211}, {4, 429}, {6, 2092}, {8, 3704}, {21, 960}, {27, 1848}, {28, 1829}, {31, 3725}, {58, 1193}, {60, 4267}, {76, 1228}, {81, 3666}, {86, 4357}, {145, 4918}, {284, 2269}, {333, 3687}, {662, 3882}, {961, 65}, {1169, 6}, {1220, 10}, {1240, 313}, {1333, 2300}, {1434, 3674}, {1474, 2354}, {1791, 72}, {1798, 3}, {2287, 3965}, {2292, 6042}, {2298, 37}, {2359, 71}, {2363, 1}, {3733, 6371}, {4267, 1682}, {4560, 3910}, {4581, 523}, {6648, 4552}, {7192, 3004}, {7199, 4509}, {8687, 4559}, {8707, 3952}

X(14535) =  X(2)X(1285)∩X(83)X(183)

Barycentrics    5*a^4 + 7*a^2*b^2 + 7*a^2*c^2 + 8*b^2*c^2 : :

X(14535) lies on these lines: {2, 1285}, {3, 6683}, {6, 9466}, {83, 183}, {99, 5024}, {194, 3329}, {381, 3589}, {1007, 7819}, {3763, 7753}, {3828, 8692}, {3830, 4045}, {3851, 7834}, {5008, 8556}, {5050, 12177}, {5055, 6033}, {5070, 6680}, {5544, 5652}, {6704, 7784}, {7735, 8367}

X(14535) = {X(11174),X(11286)}-harmonic conjugate of X(5024)

X(14536) =  ISOGONAL CONJUGATE OF X(5609)

Barycentrics   (a^8 - 5*a^6*b^2 + 9*a^4*b^4 - 7*a^2*b^6 + 2*b^8 + 3*a^6*c^2 + 4*a^2*b^4*c^2 - 7*b^6*c^2 - 8*a^4*c^4 + 9*b^4*c^4 + 3*a^2*c^6 - 5*b^2*c^6 + c^8)*(a^8 + 3*a^6*b^2 - 8*a^4*b^4 + 3*a^2*b^6 + b^8 - 5*a^6*c^2 - 5*b^6*c^2 + 9*a^4*c^4 + 4*a^2*b^2*c^4 + 9*b^4*c^4 - 7*a^2*c^6 - 7*b^2*c^6 + 2*c^8) : :
X(14536) = 9 X(477) - 4 X(10990)

X(14536) lies on the cubic K854 and these lines: {4, 12003}, {30, 14094}, {477, 10990}, {631, 9214}, {1656, 14254}

X(14536) = isogonal conjugate of X(5609)
X(14536) = X(3447)-vertex conjugate of X(5627)

X(14537) =  X(2)X(187)∩X(30)X(39)

Barycentrics    4 a^4+a^2 b^2-2 b^4+a^2 c^2+4 b^2 c^2-2 c^4 : :
X(14537) = X(2) - 3 X(598) = X(39) - 4 X(7745) = 2 X(7745) + X(7747) = X(39) + 2 X(7747) = 5 X(39) - 2 X(7756) = 10 X(7745) - X(7756) = 5 X(7753) - X(7756) = 5 X(7747) + X(7756) = 4 X(6683) - X(7802) = 2 X(3934) + X(7823) = 3 X(7812) - X(7837) = 3 X(7756) - 10 X(9300) = 3 X(39) - 4 X(9300) = 3 X(7753) - 2 X(9300) = 3 X(7745) - X(9300) = 3 X(7747) + 2 X(9300) = 3 X(9774) - X(11001) = 9 X(598) - X(11057) = X(7837) + 3 X(11361) = 3 X(4) - X(14458).

X(14537) lies on the curves Q081, K485, K487, and these lines: {2,187}, {4,5007}, {5,12815}, {6,3830}, {30,39}, {32,381}, {51,512}, {83,7842}, {115,3845}, {230,5066}, {376,2548}, {382,7772}, {384,7809}, {538,7812}, {542,5052}, {546,7755}, {547,7749}, {549,1506}, {550,9698}, {574,3534}, {626,6661}, {671,12156}, {754,8370}, {1003,7775}, {1692,5476}, {2241,11237}, {2242,11238}, {2549,14482}, {3016,3051}, {3053,5055}, {3055,11812}, {3199,7576}, {3363,13468}, {3543,5041}, {3545,7746}, {3627,7765}, {3734,7788}, {3767,3839}, {3815,6781}, {3934,7811}, {5017,11178}, {5054,5206}, {5097,6321}, {5355,12101}, {5477,8584}, {6658,7858}, {6683,7802}, {7735,10033}, {7736,9774}, {7741,9341}, {7759,14035}, {7760,14042}, {7770,7865}, {7773,7874}, {7785,7799}, {7787,7861}, {7796,14034}, {7801,14033}, {7817,12150}, {7818,11286}, {7825,7852}, {7849,7860}, {7856,14062}, {7885,7915}, {7895,7900}, {9650,10056}, {9665,10072}, {9675,13846}, {9766,11159}, {10545,11647}, {10722,14492}, {10796,13449}

X(14537) = midpoint of X(i) and X(j) for these {i,j}: {7747, 7753}, {7811, 7823}, {7812, 11361}
X(14537) = reflection of X(i) in X(j) for these {i,j}: {39, 7753}, {7753, 7745}, {7811, 3934}, {9466, 8370}
X(14537) = complement X(11057)
X(14537) = X(39)-of-reflection-triangle-of-X(6)
X(14537) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6, 3830, 11648), (187, 5475, 7603), (316, 7804, 7853), (384, 7809, 7880), (384, 7843, 7821), (3543, 7739, 7748), (3815, 6781, 8589), (3845, 5306, 115), (5475, 7737, 187), (7745, 7747, 39), (7809, 7880, 7821), (7843, 7880, 7809), (8014, 8015, 51), (12150, 14041, 7817)

leftri

Nguyen orthoimages: X(14538)-X(14541)

rightri

Let ABC be a triangle, let L be the line through B orthogonal to BC, let BA be any point on L, and let BC be the point such that BBABCC is a rectangle. Likewise, let CCBCAA be a rectangle with base CA and let AACABB be a rectangle with base AB. Let A' = ABCB∩ACBC, and define B' and C' cyclically. Let LA be the line through A' orthogonal to line BC, and define LB and LC cyclically. Nguyen Ngoc Giang found that the lines LA, LB, LC concur. See the preamble just before A(14206).

If you have The Geometer's Sketchpad, you can view Nguyen Triangle.

Let U be the ratio of the height of the rectangle BBABCC to the base; that is, U = |BAB|/a. Define V and W cyclically. Peter Moses found (September 23, 2017) that the point X' of concurrence of the lines LA, LB, LC is given by

X' = 2a2(a2 - b2 - c2)UV + 2a2(a2 - b2 - c2)UW + (a2 + b2 - c2)(a2 - b2 + c2)VW : :

If X is a point in the plane of ABC, then it has actual trilinear distances (possibly nonpositive) that are the heights of rectangles as in the above construction. Therefore, starting with X = x : y : z (barycentrics), we have U = kx/a2, where k = S/(x + y + z), and V = ky/b2 and W = kz/c2. Consequently,

X' = 2a2c2(a2 - b2 - c2)xy + 2a2b2(a2 - b2 - c2)xz + a2(a2 + b2 - c2)(a2 - b2 + c2)yz : :

X' = a2(SBSCyz - b2SAxz - c2SAxy : :

X' = isogonal conjugate of the X(3)-vertex conjugate of X
X' = reflection-in-X(3) of the isogonal conjugate of P

The triangle A'B'C' is here named the X-Nguyen triangle, and the point X' is named the Nguyen orthoimage of X.

The appearance of (i,j) in the following list means that the X(j) = Nguyen orthoimage of X(i): (1,40), (2,1350), (3,20), (4,3), (5,7691), (6,376), (7,3428), (8,10310), (9,6282), (10,3430), (13,14538), (14,14539), (15,5473), (16,5474), (17,14540), (18,14541), (20,1498), (21,14110), (24,12118), (25,6776), (28,1071), (30,110), (32,11257), (36,12119), (39,12122), (40,1490), (54,550), (55,5759), (56,944), (57,5732), (58,4297), (64,4), (65,3651), (66,378), (67,7464), (68,11413), (70,12084), (74,30), (79,11012), (80,2077), (83,5188), (84,1), (96,10625), (98,511), (99,512), (100,513), (101,514), (102,515), (103,516), (104,517), (105,518), (106,519), (107,520), (108,521), (109,522), (110,523), (111,524), (112,525), (186,12121), (187,12117), (253,10606), (254,12163), (262,3098), (265,2071), (371,12124), (372,12123), (376,11820), (378,4549), (393,10605), (476,526), (477,5663), (485,11825), (486,11824), (505,12844), (511,99), (512,98), (513,104), (514,103), (515,109), (516,101), (517,100), (518,1292), (519,1293), (520,1294), (521,1295), (522,102), (523,74), (524,1296), (525,1297), (526,477), (528,2742), (530,9202), (531,9203), (541,9060), (542,691), (543,2709), (598,8722), (675,674), (681,680), (685,9409), (689,688), (690,842), (691,690), (695,7470), (697,696), (699,698), (701,700), (703,702), (705,704), (707,706), (709,708), (711,710), (713,712), (715,714), (717,716), (719,718), (721,720), (723,722), (725,724), (727,726), (729,538), (731,730), (733,732), (735,734), (737,736), (739,536), (740,6010), (741,740), (743,742), (745,744), (747,746), (753,752), (755,754), (758,6011), (759,758), (761,760), (767,766), (769,768), (773,772), (777,776), (779,778), (781,780), (783,782), (785,784), (787,786), (789,788), (791,790), (793,792), (795,794), (797,796), (803,802), (804,2698), (805,804), (807,806), (809,808), (812,12032), (813,812), (815,814), (817,816), (819,818), (825,824), (827,826), (831,830), (833,832), (835,834), (839,838), (840,528), (841,541), (842,542), (843,543), (847,7689), (898,891), (900,953), (901,900), (907,3800), (912,13397), (915,912), (916,1305), (917,916), (919,918), (924,1300), (925,924), (926,2724), (927,926), (928,2723), (929,928), (930,1510), (931,8672), (932,4083), (933,6368), (934,3900), (935,9517), (937,9841), (945,8), (952,901), (953,952), (961,9943), (963,6361), (971,934), (972,971), (1000,6244), (1002,11495), (1093,1204), (1105,5562), (1113,2574), (1114,2575), (1126,12512), (1131,12305), (1132,12306), (1138,10620), (1141,1154), (1154,930), (1173,548), (1177,10295), (1243,7411), (1289,8673), (1290,8674), (1292,3309), (1293,3667), (1294,6000), (1295,6001), (1296,1499), (1297,1503), (1300,13754), (1301,8057), (1302,8675), (1304,9033), (1305,8676), (1308,3887), (1309,8677), (1310,8678), (1311,8679), (1320,13528), (1379,3413), (1380,3414), (1381,3307), (1382,3308), (1389,3579), (1436,2096), (1477,5853), (1490,3182), (1498,3183), (1499,111), (1503,112), (1510,1141), (1593,11821), (1676,1670), (1677,1671), (2130,3355), (2131,3348), (2137,12629), (2222,3738), (2249,8680), (2291,527), (2365,2385), (2366,2386), (2367,2387), (2368,2388), (2369,2389), (2370,2390), (2371,2391), (2372,2392), (2373,2393), (2374,8681), (2375,8682), (2378,530), (2379,531), (2380,532), (2381,533), (2382,537), (2383,539), (2384,545), (2574,1114), (2575,1113), (2687,2771), (2688,2772), (2689,2773), (2690,2774), (2691,2775), (2692,2776), (2693,2777), (2694,2778), (2695,2779), (2696,2780), (2697,2781), (2698,2782), (2699,2783), (2700,2784), (2701,2785), (2702,2786), (2703,2787), (2704,2788), (2705,2789), (2706,2790), (2707,2791), (2708,2792), (2709,2793), (2710,2794), (2711,2795), (2712,2796), (2713,2797), (2714,2798), (2715,2799), (2716,2800), (2717,2801), (2718,2802), (2719,2803), (2720,2804), (2721,2805), (2722,2806), (2723,2807), (2724,2808), (2725,2809), (2726,2810), (2727,2811), (2728,2812), (2729,2813), (2730,2814), (2731,2815), (2732,2816), (2733,2817), (2734,2818), (2735,2819), (2736,2820), (2737,2821), (2738,2822), (2739,2823), (2740,2824), (2741,2825), (2742,2826), (2743,2827), (2744,2828), (2745,2829), (2746,2830), (2747,2831), (2748,2832), (2749,2833), (2750,2834), (2751,2835), (2752,2836), (2753,2837), (2754,2838), (2755,2839), (2756,2840), (2757,2841), (2758,2842), (2759,2843), (2760,2844), (2761,2845), (2762,2846), (2763,2847), (2764,2848), (2765,2849), (2766,2850), (2767,2851), (2768,2852), (2769,2853), (2770,2854), (2771,1290), (2772,2690), (2773,2695), (2774,2688), (2775,2752), (2776,2758), (2777,1304), (2778,2766), (2779,2689), (2780,2770), (2781,935), (2782,805), (2783,2703), (2784,2702), (2785,2708), (2786,2700), (2787,2699), (2788,2711), (2789,2712), (2790,2713), (2791,2714), (2792,2701), (2793,843), (2794,2715), (2795,2704), (2796,2705), (2797,2706), (2798,2707), (2799,2710), (2800,2222), (2801,1308), (2802,2743), (2803,2744), (2804,2745), (2805,2746), (2806,2747), (2807,929), (2808,927), (2809,2736), (2810,2737), (2811,2738), (2812,2739), (2813,2740), (2814,2751), (2815,2757), (2816,2762), (2817,2765), (2818,1309), (2819,2768), (2820,2725), (2821,2726), (2822,2727), (2823,2728), (2824,2729), (2826,840), (2827,2718), (2828,2719), (2829,2720), (2830,2721), (2831,2722), (2835,2730), (2836,2691), (2841,2731), (2842,2692), (2846,2732), (2849,2733), (2850,2694), (2852,2735), (2854,2696), (2855,2869), (2856,2870), (2857,2871), (2858,2872), (2859,2873), (2860,2874), (2861,2875), (2862,2876), (2863,2877), (2864,2878), (2865,2879), (2866,2880), (2867,2881), (2868,2882), (2980,5890), (3062,3576), (3182,3353), (3183,2130), (3222,3221), (3224,12251), (3307,1382), (3308,1381), (3309,105), (3333,12120), (3345,84), (3346,64), (3347,3345), (3348,3346), (3353,3472), (3354,3347), (3406,9821), (3413,1380), (3414,1379), (3424,6), (3426,2), (3427,55), (3429,58), (3431,3534), (3432,12254), (3440,6770), (3441,6773), (3445,12245), (3446,13199), (3447,12383), (3459,12307), (3473,3354), (3521,7488), (3527,3522), (3531,10304), (3532,3529), (3563,3564), (3564,3565), (3565,3566), (3566,3563), (3577,165), (3637,2131), (3657,7429), (3667,106), (3738,2716), (3849,6233), (3887,2717), (3900,972), (3906,14388), (4588,4777), (4846,22), (5545,4843), (5553,11249), (5606,8702), (5627,12041), (5663,476), (5840,6099), (5962,13496), (5966,5965), (5970,5969), (6000,107), (6001,108), (6002,741), (6003,759), (6010,6002), (6011,6003), (6012,6004), (6013,6005), (6014,6006), (6015,6007), (6016,6008), (6017,6009), (6078,6084), (6079,6085), (6080,6086), (6081,6087), (6082,6088), (6083,6089), (6088,6093), (6135,6364), (6136,6365), (6145,3520), (6183,6182), (6200,13666), (6233,8704), (6323,3849), (6325,8705), (6380,6379), (6396,13786), (6401,12223), (6402,12224), (6502,8984), (6551,6550), (6571,8710), (6573,8711), (6574,8712), (6575,8713), (6577,8714), (6578,6367), (6662,11440), (7350,1742), (7953,7927), (7954,7950), (8057,5897), (8059,8058), (8064,8063), (8599,9142), (8652,4802), (8674,2687), (8676,917), (8677,2734), (8684,3808), (8685,3810), (8686,3880), (8687,3910), (8690,4139), (8691,4160), (8693,4762), (8694,4778), (8695,4844), (8696,4912), (8697,4926), (8698,4943), (8699,4962), (8700,4971), (8701,4977), (8702,5951), (8704,6323), (8705,6236), (8706,6363), (8707,6371), (8708,6372), (8709,6373), (8884,185), (8917,11372), (8946,12256), (8948,12257), (9003,841), (9033,2693), (9056,8999), (9057,9000), (9058,9001), (9059,9002), (9060,9003), (9061,9004), (9062,9005), (9063,9006), (9064,9007), (9065,9008), (9066,9009), (9067,9010), (9068,9011), (9069,9012), (9070,9013), (9071,9014), (9072,9015), (9073,9016), (9074,9017), (9075,9018), (9076,9019), (9077,9020), (9078,9021), (9079,9022), (9080,9023), (9081,9024), (9082,9025), (9083,9026), (9084,9027), (9085,9028), (9086,9029), (9087,9030), (9088,9031), (9089,9032), (9090,9034), (9091,9035), (9092,9036), (9093,9037), (9094,9038), (9095,9039), (9096,9040), (9097,9041), (9098,9042), (9099,9043), (9100,9044), (9101,9045), (9102,9046), (9103,9047), (9104,9048), (9105,9049), (9106,9050), (9107,9051), (9108,9052), (9109,9053), (9110,9054), (9111,9055), (9141,9145), (9150,888), (9217,13172), (9517,2697), (9518,2741), (9519,2748), (9520,2749), (9521,2750), (9522,2753), (9523,2754), (9524,2755), (9525,2756), (9526,2759), (9527,2760), (9528,2761), (9529,2763), (9530,2764), (9531,2767), (9532,2769), (9830,13241), (9831,9830), (10152,12096), (10293,23), (10307,999), (10308,1385), (10309,56), (11270,1657), (11559,3153), (11593,11594), (11636,3906), (11645,11636), (11738,381), (11744,186), (11816,6102), (12074,12073), (13452,382), (13478,991), (13489,5876), (13573,2935), (13597,13391), (13603,549), (13754,925), (14074,14077), (14228,3311), (14229,9732), (14232,371), (14234,9738), (14236,7692), (14237,372), (14238,9739), (14240,7690), (14243,3312), (14244,9733), (14370,12252), (14388,11645), (14458,182), (14483,8703), (14490,3524)

The mapping X → X' takes the circumcircle to the infinity line, and the infinity line to the circumcircle, in cycles. In the following list, i → j means that X(i) → X(j):

30 → 110 → 523 → 74 → 30
100 → 513 → 104 → 517 → 100
101 → 514 → 103 → 516 → 101
102 → 515 → 109 → 512 → 102
105 → 518 → 1292 → 3309 → 105
106 → 519 → 12193 → 3667 → 106
107 → 520 → 1294 → 6000 → 107
108 → 521 → 1295 → 6001 → 108

Chains of Nguyen orthoimages can be read from the list. These include the following: 3426 → 2 → 1350
3424 → 6 → 376 → 11820
3637 → 2131 → 3348 → 3346 → 64 → 4 → 3 → 20 → 1498 → 3183 → 2130 → 3355
3473 → 3354 → 3347 → 3345 → 84 → 1 → 40 → 1490 → 3182 → 3353 → 3472

All the points on those last two chains lie on the Darboux cubic, K004. p

The inverse of the mapping P → P', where P = p : q : r, is given by

a^2 (2 a^2 b^2 p-2 b^4 p+2 b^2 c^2 p+a^4 q-b^4 q-2 a^2 c^2 q+c^4 q+2 a^2 b^2 r-2 b^4 r+2 b^2 c^2 r) (2 a^2 c^2 p+2 b^2 c^2 p-2 c^4 p+2 a^2 c^2 q+2 b^2 c^2 q-2 c^4 q+a^4 r-2 a^2 b^2 r+b^4 r-c^4 r) : : ,

this being the isogonal conjugate of the reflection-in-X(3) of P. (Peter Moses, September 25, 2017)

The X-Nguyen triangle A'B'C' is perspective to ABC if X is on one of the following: (1) the line at infinity; (2), the circumcircle; (3) a quartic curve that passes through X(i) for i = 3, 4, 64, 2574, 2575; and (4) the circle with center X(3) and radius R*sqrt(5), here named the Nguyen-Moses circle. If X is on the circumcircle, then the points given by combos 4*X(3) - (5 + sqrt(5))*X and 4*X(3) - (5 - sqrt(5))*X is on the Nguyen-Moses circle. (Peter Moses, September 25, 2017)

X(14538) =  REFLECTION OF X(15) IN X(3)

Barycentrics    (SB+SC) (-SB SC+SA (Sqrt(3) S+SA+SW)) : :

X(14538) lies on these lines: {2, 7684}, {3, 6}, {4, 623}, {20, 621}, {23, 11131}, {30, 5463}, {74, 9202}, {98, 6582}, {147, 616}, {316, 11133}, {323, 14170}, {325, 9749}, {376, 531}, {383, 3643}, {532, 6770}, {618, 1080}, {631, 6671}, {842, 9203}, {1296, 2378}, {1297, 5995}, {2379, 2709}, {2710, 5994}, {2780, 9162}, {3130, 5651}, {3132, 3917}, {3576, 11707}, {5663, 13859}, {5980, 5999}, {5981, 6194}, {7492, 14169}, {7685, 11304}, {11003, 11126}

X(14538) =midpoint of X(20) and X(621)
X(14538) = reflection of X(i) in X(j) for these {i,j}: {4, 623}, {15, 3}, {1080, 618}, {2080, 13349}, {5611, 13350}
X(14538) = Nguyen orthoimage of X(13)
X(14538) = circumtangential-isogonal conjugate of X(33388)
X(14538) ={X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,1350,14539), (3, 5611, 13350), (3, 5615, 182), (3, 5864, 61), (3, 5865, 5238), (3, 9732, 3389), (3, 9733, 3390), (3, 9736, 10646), (3, 11486, 5085), (16, 3106, 62), (182, 5615, 62), (3098, 9736, 3), (5611, 13350, 15)
X(14538) = anticomplement X(7684)

X(14539) =  REFLECTION OF X(16) IN X(3)

Barycentrics    (SB+SC) (-SB SC+SA (Sqrt(3) S+SA+SW)) : :(SB+SC) (-SB SC+SA (-Sqrt(3) S+SA+SW)) : :

X)14539) lies on these lines: {2, 7685}, {3, 6}, {4, 624}, {20, 622}, {23, 11130}, {30, 5464}, {74, 9203}, {98, 6295}, {147, 617}, {316, 11132}, {323, 14169}, {325, 9750}, {376, 530}, {383, 619}, {533, 6773}, {631, 6672}, {842, 9202}, {1080, 3642}, {1296, 2379}, {1297, 5994}, {2378, 2709}, {2710, 5995}, {2780, 9163}, {3129, 5651}, {3131, 3917}, {3576, 11708}, {5663, 13858}, {5980, 6194}, {5981, 5999}, {7492, 14170}, {7684, 11303}, {11003, 11127}

X(14539) = midpoint of X(20) and X(622)
X(14539) = reflection of X(i) in X(j) for these {i,j}: {4, 624}, {16, 3}, {383, 619}, {2080, 13350}, {5615, 13349}
X(14539) = anticomplement X(7685)
X(14539) = circumtangential-isogonal conjugate of X(33389)
X(14539) = Nguyen orthoimage of X(14)
X(14539) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,1350,14538), (3, 5611, 13350), (3, 5615, 182), (3, 5864, 61), (3, 5865, 5238), (3, 9732, 3389), (3, 9733, 3390), (3, 9736, 10646), (3, 11486, 5085), (16, 3106, 62), (182, 5615, 62), (3098, 9736, 3), (5611, 13350, 15)

X(14540) =  REFLECTION OF X(61) IN X(3)

Barycentrics    (SB+SC) (S SA+Sqrt(3) (SA^2-SB SC+SA SW)) : :
X(14540) = 5 X(631) - 4 X(6694)

X(14540) lies on these lines: {3, 6}, {4, 635}, {20, 616}, {376, 533}, {383, 636}, {550, 5474}, {631, 6694}, {3130, 3917}, {6636, 11126}, {7768, 11128}

X(14540) = midpoint of X(20)) and X(633)
X(14540) = reflection of X(i) in X(j) for these {i,j}: {4, 635}, {61, 3}
X(14540) = circumtangential-isogonal conjugate of X(33421)
X(14540) = Nguyen orthoimage of X(17)
X(14540) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,1350,14541), (3, 5864, 62), (3, 5865, 15)

X(14541) =  REFLECTION OF X(62) IN X(3)

Barycentrics    (SB+SC) (S SA-Sqrt(3) (SA^2-SB SC+SA SW)) : :
X(14541) = 5 X(631) - 4 X(6695)

X(14541) lies on these lines: {3, 6}, {4, 636}, {20, 617}, {376, 532}, {550, 5473}, {631, 6695}, {635, 1080}, {3129, 3917}, {6636, 11127}, {7768, 11129}

X(14541) = midpoint X(20) and X(634)
X(14541) = reflection of X(i) in X(j) for these {i,j}: {4, 636}, {62, 3}
X(14541) = circumtangential-isogonal conjugate of X(33420)
X(14541) = Nguyen orthoimage of X(18)
X(14541) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,1350,14540), (3, 5865, 61), (3, 5864, 16)

X(14542) =  PERSPECTOR OF THESE TRIANGLES: ABC AND X(4)-NGUYEN

Barycentrics    (a^8+2 a^6 b^2-6 a^4 b^4+2 a^2 b^6+b^8-2 a^6 c^2-2 a^4 b^2 c^2-2 a^2 b^4 c^2-2 b^6 c^2-2 a^2 b^2 c^4+2 a^2 c^6+2 b^2 c^6-c^8) (a^8-2 a^6 b^2+2 a^2 b^6-b^8+2 a^6 c^2-2 a^4 b^2 c^2-2 a^2 b^4 c^2+2 b^6 c^2-6 a^4 c^4-2 a^2 b^2 c^4+2 a^2 c^6-2 b^2 c^6+c^8) : :

X(14542) lies on these lines: {3,12233}, {6,3575}, {20,1176}, {51,14457}, {64,427}, {65,11393}, {66,185}, {67,13148}, {68,389}, {69,5889}, {70,5890}, {74,3541}, {265,11746}, {1173,12289}, {1177,11424}, {1192,7499}, {1899,6145}, {3313,10996}, {3426,5878}, {3527,12241}, {7399,9786}, {9815,13567}

X(14542) = isogonal conjugate of X(7503)

X(14543) =  ANTICOMPLEMENT OF X(4466)

Barycentrics    (a - b)*(a - c)*(2*a^3 + a^2*b + b^3 + a^2*c - b^2*c - b*c^2 + c^3) : :

X(14543) lies on these lines: {2, 1762}, {7, 1781}, {20, 2822}, {100, 190}, {101, 1305}, {107, 110}, {144, 1761}, {191, 6172}, {523, 7479}, {651, 653}, {666, 4581}, {692, 2398}, {846, 5281}, {857, 7359}, {874, 7258}, {1944, 7291}, {2173, 8680}, {2287, 11683}, {2407, 4556}, {2948, 9803}, {4608, 4629}, {4858, 5773}, {8756, 9028}

X(14543) = reflection of X(857) in X(7359)
X(14543) = cevapoint of X(514) and X(6678)
X(14543) = crosspoint of X(i) and X(j) for these (i,j): {99, 658}, {190, 648}
X(14543) = trilinear pole of line X(440)X(950)
X(14543) = crossdifference of every pair of points on line X(1015)X(3269)
X(14543) = crosssum of X(i) and X(j) for these (i,j): {512, 657}, {647, 649}
X(14543) = anticomplement X(4466)
X(14543) = X(i)-aleph conjugate of X(j) for these (i,j): {190, 20}, {651, 3216}, {653, 1714}, {664, 2}, {4551, 1045}, {4552, 191}, {4998, 3882}, {14087, 20}
X(14543) = X(190)-daleth conjugate of X(13589)
X(14543) = X(i)-zayin conjugate of X(j) for these (i,j): {72, 649}, {1215, 4581}, {1427, 657}, {1441, 514}
X(14543) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {59, 2897}, {100, 13219}, {112, 149}, {162, 150}, {250, 75}, {1110, 3151}, {1783, 3448}, {2149, 3152}, {4567, 1370}, {4570, 4329}, {5379, 69}, {7012, 2893}, {7115, 2475}
X(14543) = isoconjugate of X(j) and X(j) for these (i,j): {513, 2983}, {649, 1257}, {650, 951}
X(14543) = {X(651),X(653)}-harmonic conjugate of X(4566)
X(14543) = barycentric product X(i)*X(j) for these {i,j}: {99, 1834}, {440, 648}, {664, 950}, {668, 1104}, {1842, 4561}, {2264, 4554}
X(14543) = barycentric quotient X(i)/X(j) for these {i,j}: {100, 1257}, {101, 2983}, {109, 951}, {440, 525}, {950, 522}, {1104, 513}, {1834, 523}, {1842, 7649}, {2264, 650}

X(14544) =  X(100)X(658)∩X(107)X(110)

Barycentrics    (-a + b)*(a - c)*(2*a^4 + a^3*b - a^2*b^2 - a*b^3 - b^4 + a^3*c + 2*a^2*b*c + a*b^2*c - a^2*c^2 + a*b*c^2 + 2*b^2*c^2 - a*c^3 - c^4) : :

X(14544) lies on these lines: {20, 2816}, {100, 658}, {107, 110}, {643, 4427}, {651, 1897}, {1068, 5906}, {1331, 4552}, {1332, 3952}, {2407, 4636}, {3241, 6264}

X(14544) = crosssum of X(647) and X(663)
X(14544) = crosspoint of X(648) and X(664)
X(14544) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {108, 3448}, {250, 3869}, {651, 13219}, {1262, 2897}, {2149, 3151}, {5379, 3436}, {7012, 1330}, {7115, 2895}, {7128, 2893}
X(14544) = trilinear pole of line X(1901)X(4292)
X(14544) = barycentric product X(i)*X(j) for these {i,j}: {99, 1901}, {190, 4292}
X(14544) = barycentric quotient X(i)/X(j) for these {i,j}: {1901, 523}, {4292, 514}

X(14545) =  X(101)X(514)∩X(107)X(110)

Barycentrics    (-a + b)*(a - c)*(a^8 - a^7*b - a^6*b^2 + a^5*b^3 - a^4*b^4 + a^3*b^5 + a^2*b^6 - a*b^7 - a^7*c - a^6*b*c + 3*a^5*b^2*c + a^4*b^3*c - 3*a^3*b^4*c + a^2*b^5*c + a*b^6*c - b^7*c - a^6*c^2 + 3*a^5*b*c^2 + 8*a^4*b^2*c^2 + 2*a^3*b^3*c^2 - a^2*b^4*c^2 + 3*a*b^5*c^2 + 2*b^6*c^2 + a^5*c^3 + a^4*b*c^3 + 2*a^3*b^2*c^3 - 2*a^2*b^3*c^3 - 3*a*b^4*c^3 + b^5*c^3 - a^4*c^4 - 3*a^3*b*c^4 - a^2*b^2*c^4 - 3*a*b^3*c^4 - 4*b^4*c^4 + a^3*c^5 + a^2*b*c^5 + 3*a*b^2*c^5 + b^3*c^5 + a^2*c^6 + a*b*c^6 + 2*b^2*c^6 - a*c^7 - b*c^7) : :

X(14545) lies on these lines: {101, 514}, {107, 110}, {651, 13149}

X(14546) =  X(107)X(110)∩X(109)X(190)

Barycentrics    (-a + b)*(a - c)*(a^6 + a^5*b - a^2*b^4 - a*b^5 + a^5*c + 5*a^4*b*c + 2*a^3*b^2*c + a*b^4*c - b^5*c + 2*a^3*b*c^2 + 2*a^2*b^2*c^2 + 2*b^3*c^3 - a^2*c^4 + a*b*c^4 - a*c^5 - b*c^5) : :

X(14546) lies on these lines: {107, 110}, {109, 190}, {651, 4391}

X(14546) = the root of cubic K385; see Bernard Gibert, K385.

X(14547) =  X(1)X(4)∩X(6)X(31)

Barycentrics    a^2*(a - b - c)*(a^2*b - b^3 + a^2*c + 2*a*b*c + b^2*c + b*c^2 - c^3) : :

X(14547) lies on the cubic K382 and these lines: {1, 4}, {2, 1818}, {3, 1451}, {6, 31}, {9, 2318}, {11, 3136}, {25, 48}, {35, 580}, {36, 7430}, {37, 1864}, {41, 5452}, {43, 5218}, {51, 228}, {57, 955}, {65, 4300}, {81, 1936}, {181, 2223}, {184, 2317}, {200, 3686}, {204, 3192}, {210, 1212}, {219, 13615}, {221, 6254}, {284, 2189}, {354, 1427}, {375, 4557}, {386, 1453}, {405, 3682}, {464, 1040}, {500, 942}, {572, 5285}, {601, 3215}, {610, 5338}, {663, 11193}, {750, 11502}, {846, 1776}, {899, 5432}, {943, 3074}, {995, 13384}, {1037, 3477}, {1104, 1193}, {1214, 5728}, {1400, 2352}, {1449, 7070}, {1450, 3576}, {1465, 11018}, {1486, 2187}, {1497, 10267}, {1742, 3474}, {1824, 1953}, {1841, 1859}, {1858, 2292}, {1962, 2310}, {2173, 2355}, {2188, 7151}, {2262, 3198}, {2304, 2333}, {2323, 2328}, {2667, 11997}, {2999, 10383}, {3000, 11246}, {3100, 3151}, {3191, 12572}, {3240, 5281}, {3295, 7078}, {3304, 4322}, {3666, 7004}, {3931, 12711}, {4306, 11518}, {4337, 5902}, {4343, 14100}, {4551, 13405}, {5287, 9817}, {5320, 6056}, {5453, 12433}, {7073, 9629}, {10459, 10950}

X(14547) = X(i)-Ceva conjugate of X(j) for these (i,j): {1, 2294}, {162, 652}, {942, 2260}, {4551, 650}, {6011, 649}
X(14547) = crosspoint of X(i) and X(j) for these (i,j): {1, 284}, {9, 29}, {55, 7073}
X(14547) = crossdifference of every pair of points on line {514, 652}
X(14547) = crosssum of X(i) and X(j) for these (i,j): {1, 226}, {7, 1442}, {57, 73}, {943, 2982}
X(14547) = X(100)-beth conjugate of X(13405)
X(14547) = X(i)-isoconjugate of X(j) for these (i,j): {2, 2982}, {7, 943}, {85, 2259}, {273, 1794}, {1175, 1441}
X(14547) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 581, 73), (1, 950, 2654), (1, 1064, 1457), (1, 1745, 3487), (1, 10382, 33), (6, 55, 212), (9, 3190, 2318), (37, 1864, 7069), (42, 2293, 55), (51, 228, 2183), (55, 11435, 71), (500, 942, 4303), (3666, 10391, 7004), (5256, 7675, 1040)
X(14547) = barycentric product X(i)*X(j) for these {i,j}: {6, 6734}, {8, 2260}, {9, 942}, {21, 2294}, {55, 5249}, {63, 1859}, {78, 1841}, {219, 1838}, {226, 8021}, {281, 4303}, {283, 1865}, {284, 442}, {445, 8606}, {500, 7110}
X(14547) = barycentric quotient X(i)/X(j) for these {i,j}: {31, 2982}, {41, 943}, {442, 349}, {942, 85}, {1838, 331}, {1841, 273}, {1859, 92}, {2175, 2259}, {2260, 7}, {2294, 1441}, {4303, 348}, {5249, 6063}, {6734, 76}, {8021, 333}

X(14548) =  X(1)X(348)∩X(2)X(6)

Barycentrics    -a^4 + 2*a^3*b - 2*a*b^3 + b^4 + 2*a^3*c + 6*a^2*b*c + 2*a*b^2*c - 2*b^3*c + 2*a*b*c^2 + 2*b^2*c^2 - 2*a*c^3 - 2*b*c^3 + c^4 : :

X(14548) lies on the cubic K382 and these lines: {1, 348}, {2, 6}, {4, 14520}, {7, 354}, {20, 1434}, {85, 938}, {150, 1056}, {200, 3879}, {326, 3870}, {1014, 7411}, {1440, 2192}, {3475, 7179}, {3486, 7176}, {3488, 5088}, {3664, 11019}, {3674, 11518}, {3726, 4419}, {3745, 10578}, {4340, 13727}, {4357, 10582}, {4644, 10025}, {4847, 10436}, {4955, 12701}, {5269, 8817}, {6738, 9312}, {6744, 10481}, {7247, 11037}, {7278, 10573}
{X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3945, 5738, 69)

X(14548) = barycentric product X(85)*X(7675)
X(14548) = barycentric quotient X(7675)/X(9)

X(14549) =  X(4)X(991)∩X(37)X(38)

Barycentrics    a*(a*b - b^2 + a*c + b*c)*(a*b + a*c + b*c - c^2)*(a^3 - a*b^2 - 2*a*b*c - 2*b^2*c - a*c^2 - 2*b*c^2) : :

X(14549) lies on the cubic K382 and these lines: {4, 991}, {37, 38}

X(14549) = barycentric product X(5271)*X(13476)
X(14549) = barycentric quotient X(i)/X(j) for these {i,j}: {5295, 4043}, {5320, 4251}

X(14550) =  X(2)X(40)∩X(6)X(84)

Barycentrics    a*(a^3 + 3*a^2*b + 3*a*b^2 + b^3 + a^2*c - 2*a*b*c + b^2*c - a*c^2 - b*c^2 - c^3)*(a^3 + a^2*b - a*b^2 - b^3 + 3*a^2*c - 2*a*b*c - b^2*c + 3*a*c^2 + b*c^2 + c^3)*(a^3 - a^2*b - a*b^2 + b^3 - a^2*c + 2*a*b*c + 3*b^2*c - a*c^2 + 3*b*c^2 + c^3) : :

X(14550) lies on the cubic K384 and these lines: {2, 40}, {6, 84}

X(14551) =  X(2)X(84)∩X(6)X(40)

Barycentrics    a*(a^3 + a^2*b - a*b^2 - b^3 + a^2*c + 2*a*b*c - 3*b^2*c - a*c^2 - 3*b*c^2 - c^3)*(a^3 + 3*a^2*b + 3*a*b^2 + b^3 - a^2*c + 2*a*b*c - b^2*c - a*c^2 - b*c^2 + c^3)*(a^3 - a^2*b - a*b^2 + b^3 + 3*a^2*c + 2*a*b*c - b^2*c + 3*a*c^2 - b*c^2 + c^3) : :

X(14551) lies on the cubic K384 and these lines: on lines {2, 84}, {6, 40}, {963, 2910}

X(14552) =  X(2)X(6)∩X(8)X(20)

Barycentrics    -3*a^3 - a^2*b + 3*a*b^2 + b^3 - a^2*c + 2*a*b*c + 3*b^2*c + 3*a*c^2 + 3*b*c^2 + c^3 : :
X(14552) = 3 X(2) - 4 X(5737)

X(14552) lies on the cubic K384 and these lines: {2, 6}, {7, 4001}, {8, 20}, {10, 4340}, {57, 3686}, {75, 9965}, {85, 4359}, {144, 321}, {165, 4061}, {200, 991}, {239, 4352}, {253, 10432}, {286, 6994}, {306, 5273}, {319, 345}, {329, 4416}, {346, 3219}, {347, 1943}, {452, 10449}, {1330, 5177}, {1792, 4189}, {2321, 3929}, {2345, 4641}, {2550, 4042}, {3187, 3672}, {3332, 4847}, {3421, 5774}, {3474, 3696}, {3617, 7270}, {3666, 5839}, {3681, 7172}, {3687, 5744}, {3706, 5698}, {3707, 7308}, {3926, 7058}, {3928, 4034}, {3974, 5220}, {4101, 5703}, {4384, 9776}, {5231, 5733}, {5287, 5296}, {5686, 10327}, {6904, 9534}

X(14552) = reflection of X(5712) in X(5737)
X(14552) = anticomplement X(5712)
X(14552) = barycentric product X(69)*X(7498)
X(14552) = barycentric quotient X(7498)/X(4)
X(14552) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (69, 333, 2), (940, 966, 2), (1150, 3578, 5739), (1150, 5739, 2), (2895, 5361, 2), (4001, 5271, 7), (4416, 11679, 329), (5712, 5737, 2)

X(14553) =  X(2)X(1901)∩X(37)X(40)

Barycentrics    a^2*(a^3 + 3*a^2*b + 3*a*b^2 + b^3 + 3*a^2*c + 2*a*b*c + 3*b^2*c - a*c^2 - b*c^2 - 3*c^3)*(a^3 + 3*a^2*b - a*b^2 - 3*b^3 + 3*a^2*c + 2*a*b*c - b^2*c + 3*a*c^2 + 3*b*c^2 + c^3) : :

X(14553) lies on the cubic K384, the hyperbola {A,B,C,X(12),X(6)}, and these lines: {2, 1901}, {6, 2360}, {37, 40}, {42, 198}, {208, 1880}, {221, 1400}, {2305, 8770}

X(14553) = isoconjugate of X(63) and X(7498)
X(14553) = barycentric quotient X(25)/X(7498)

X(14554) =  X(2)X(3882)∩X(10)X(11)

Barycentrics    (a^2*b - b^3 + a^2*c - 3*a*b*c + a*c^2 + b*c^2)*(a^2*b + a*b^2 + a^2*c - 3*a*b*c + b^2*c - c^3) : :

X(14554) lies on the cubic K387, the Kiepert hyperbola, and these lines: {2, 3882}, {4, 3216}, {10, 11}, {76, 5233}, {226, 1086}, {321, 3452}, {662, 14534}, {899, 13576}, {908, 1266}, {946, 3987}, {1193, 4551}, {2183, 3911}, {3687, 4033}, {4049, 4927}, {4383, 13478}, {5397, 6946}

X(14554) = isogonal conjugate of X(5053)
X(14554) = X(4695)-cross conjugate of X(75)
X(14554) = cevapoint of X(i) and X(j) for these (i,j): {661, 1647}, {900, 2170}, {1193, 2183}
X(14554) = trilinear pole of line {523, 2292, 6615}

X(14555) =  X(1)X(4104)∩X(2)X(6)

Barycentrics    (a - b - c)*(a^2 + 2*a*b + b^2 + 2*a*c + c^2) : :

X(14555) lies on these lines: {1, 4104}, {2, 6}, {4, 970}, {8, 210}, {9, 345}, {11, 4042}, {55, 4023}, {57, 4416}, {63, 2183}, {75, 329}, {200, 3883}, {223, 348}, {226, 4384}, {306, 344}, {320, 9776}, {386, 13725}, {390, 3996}, {443, 1330}, {452, 1043}, {469, 2322}, {631, 13323}, {908, 5271}, {936, 937}, {957, 3421}, {1444, 11350}, {1792, 11344}, {1999, 5839}, {2051, 5816}, {2348, 5273}, {2550, 4388}, {2999, 4357}, {3091, 5799}, {3210, 4419}, {3306, 4001}, {3416, 3740}, {3434, 4651}, {3452, 3686}, {3617, 5835}, {3699, 7172}, {3703, 3715}, {3707, 5745}, {3711, 4030}, {3752, 4643}, {3789, 10453}, {3820, 5774}, {3875, 4656}, {3886, 4061}, {3912, 7308}, {3926, 11343}, {4035, 6666}, {4046, 4387}, {4359, 5905}, {4361, 4415}, {4684, 10582}, {4734, 9791}, {4966, 8167}, {5044, 5814}, {5084, 10449}, {5268, 5847}, {5943, 10477}, {6838, 12324}, {6863, 11411}, {6959, 11487}, {8896, 9816}

X(14555) = anticomplement of X(37674)
X(14555) = X(4254)-cross conjugate of X(4194)
X(14555) = X(7256)-beth conjugate of X(7172)
X(14555) = X(1444)-gimel conjugate of X(11427)
X(14555) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 193, 940), (2, 391, 333), (2, 5739, 69), (6, 5743, 2), (8, 2899, 3714), (8, 3876, 1265), (9, 3687, 345), (210, 3966, 8), (306, 3305, 344), (312, 4886, 8), (333, 5233, 2), (940, 5241, 2), (1211, 4383, 2), (3452, 3686, 11679), (4113, 4863, 8), (5278, 5739, 11433), (5278, 5741, 2)
X(14555) = barycentric product X(i)*X(j) for these {i,j}: {69, 4194}, {75, 5250}, {76, 4254}, {312, 5256}, {314, 3931}, {3718, 7713}
X(14555) = barycentric quotient X(i)/X(j) for these {i,j}: {3931, 65}, {4194, 4}, {4254, 6}, {5250, 1}, {5256, 57}, {7713, 34}

X(14556) =  X(8)X(392)∩X(57)X(2183)

Barycentrics    (a^2 - 4*a*b + b^2 - c^2)*(a^2 - b^2 - 4*a*c + c^2)*(a^2 + 2*a*b + b^2 + 2*a*c - 2*b*c + c^2) : :

X(14556) lies on the cubic K387 and these lines: {8, 392}, {57, 2183}

X(14556) = X(i)-isoconjugate of X(j) for these (i,j): {999, 2297}, {3306, 7050}
X(14556) = barycentric product X(1000)*X(3672)
X(14556) = barycentric quotient X(i)/X(j) for these {i,j}: {1000, 1219}, {1191, 999}, {1697, 3872}, {2999, 3306}, {4646, 3753}, {4656, 4054}

X(14557) =  X(4)X(8)∩X(6)X(57)

Barycentrics    a*(a^4*b + 2*a^3*b^2 - 2*a*b^4 - b^5 + a^4*c - 2*a^2*b^2*c + b^4*c + 2*a^3*c^2 - 2*a^2*b*c^2 + 4*a*b^2*c^2 - 2*a*c^4 + b*c^4 - c^5) : :

X(14557) lies on the cubic K387 and these lines: {4, 8}, {6, 57}, {44, 1726}, {51, 5728}, {209, 2835}, {226, 2262}, {389, 1071}, {999, 13737}, {1211, 3452}, {1214, 2183}, {2093, 3987}, {3057, 4656}, {3666, 4266}, {5908, 6848}, {6244, 7085}, {7078, 7713}

X(14557) = {X(223),X(2270)}-harmonic conjugate of X(11347)

X(14558) =  X(3)X(7768)∩X(22)X(316)

Barycentrics    a^2*(a^8 - 2*a^6*b^2 + 2*a^2*b^6 - b^8 - 2*a^6*c^2 + a^4*b^2*c^2 + 2*a^2*b^4*c^2 - b^6*c^2 + 2*a^2*b^2*c^4 + b^4*c^4 + 2*a^2*c^6 - b^2*c^6 - c^8) : :

X(14558) lies on the cubic K377 and these lines: {3, 7768}, {22, 316}, {305, 7771}, {2070, 7752}, {2071, 7782}, {6636, 11057}, {6644, 7769}, {7488, 7814}, {7502, 7809}, {7512, 7860}, {7763, 10298}

X(14559) =  ISOGONAL CONJUGATE OF X(9213)

Barycentrics    (a - b)*(a + b)*(a - c)*(a + c)*(2*a^2 - b^2 - c^2)*(a^2 - a*b + b^2 - c^2)*(a^2 + a*b + b^2 - c^2)*(a^2 - b^2 - a*c + c^2)*(a^2 - b^2 + a*c + c^2) : :

X(14559) lies on the cubics K064 and K396 and these lines: {6, 13}, {110, 476}, {526, 2407}, {690, 5467}, {5609, 14254}, {5642, 9717}, {5967, 6593}, {9143, 9214}

X(14559) = midpoint of X(9143) and X(9214)
X(14559) = reflection of X(i) in X(j) for these {i,j}: {265, 14356}, {5967, 6593}
X(14559) = isogonal conjugate of X(9213)
X(14559) = X(14559) = X(2407)-line conjugate of X(526)
X(14559) = cevapoint of X(690) and X(5642)
X(14559) = trilinear pole of line {187, 1648}
X(14559) = crossdifference of every pair of points on line {526, 2088}
X(14559) = X(i)-isoconjugate of X(j) for these (i,j): {1, 9213}, {526, 897}, {671, 2624}, {923, 3268}, {5466, 6149}
X(14559) = barycentric product X(i)X(j) for these {i,j}: {94, 5467}, {265, 4235}, {476, 524}, {1989, 5468}
X(14559) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 9213}, {187, 526}, {351, 2088}, {476, 671}, {524, 3268}, {922, 2624}, {1989, 5466}, {3292, 8552}, {4235, 340}, {5467, 323}, {5468, 7799}, {5642, 5664}, {11060, 9178}

X(14560) =  ISOGONAL CONJUGATE OF X(3268)

Barycentrics    a^2*(a - b)*(a + b)*(a - c)*(a + c)*(a^2 - a*b + b^2 - c^2)*(a^2 + a*b + b^2 - c^2)*(a^2 - b^2 - a*c + c^2)*(a^2 - b^2 + a*c + c^2) : :

X(14560) lies on the cubics K396 and these lines: {110, 476}, {182, 14356}, {250, 1510}, {265, 1177}, {512, 1576}, {685, 10412}, {692, 4705}, {1495, 3003}, {1692, 11060}, {1976, 1989}, {2492, 2715}, {5627, 14157}, {9178, 9206}

X(14560) = isogonal conjugate of X(3268)
X(14560) = X(14398)-cross conjugate of X(6)
X(14560) = X(5994)-Hirst inverse of X(5995)
X(14560) = X(i)-vertex conjugate of X(j) for these (i,j): {2966, 5649}, {5467, 9186}, {5649, 2966}, {9186, 5467}
X(14560) = cevapoint of X(i) and X(j) for these (i,j): {184, 9409}, {512, 1495}
X(14560) = trilinear pole of line {32, 3124}
X(14560) = polar conjugate of isotomic conjugate of X(32662)
X(14560) = crossdifference of every pair of points on line {2088, 5664}
X(14560) = crosssum of X(i) and X(j) for these (i,j): {526, 8552}, {6334, 9033}
X(14560) = X(i)-zayin conjugate of X(j) for these (i,j): {1, 3268}, {1759, 2624}
X(14560) = barycentric product of circumcircle intercepts of the Fermat axis
X(14560) = vertex conjugate of MacBeath circumconic intercepts of Brocard axis
X(14560) = X(i)-isoconjugate of X(j) for these (i,j): {1, 3268}, {75, 526}, {76, 2624}, {92, 8552}, {186, 14208}, {323, 1577}, {340, 656}, {561, 14270}, {661, 7799}, {758, 4467}, {799, 2088}, {850, 6149}, {1109, 10411}, {1273, 2616}, {2349, 5664}, {3218, 7265}, {3219, 4707}, {3678, 4453}, {3960, 3969}, {4585, 8287}, {9213, 14210}
X(14560) = barycentric product X(i)X(j) for these {i,j}: {6, 476}, {13, 5994}, {14, 5995}, {94, 1576}, {99, 11060}, {110, 1989}, {112, 265}, {163, 2166}, {477, 2437}, {1141, 1625}, {2161, 13486}, {2420, 5627}, {2715, 14356}, {4240, 11079}, {14147, 14158}
X(14560) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 3268}, {32, 526}, {110, 7799}, {112, 340}, {184, 8552}, {265, 3267}, {476, 76}, {560, 2624}, {669, 2088}, {1495, 5664}, {1501, 14270}, {1576, 323}, {1625, 1273}, {1989, 850}, {2420, 6148}, {5994, 298}, {5995, 299}, {6186, 4707}, {6187, 7265}, {11060, 523}, {14398, 3258}

X(14561) =  X(2)X(51)∩X(5)X(6)

Barycentrics    a^6 - 3*a^4*b^2 + a^2*b^4 + b^6 - 3*a^4*c^2 - 6*a^2*b^2*c^2 - b^4*c^2 + a^2*c^4 - b^2*c^4 + c^6 : :
X(14561) = X(3) + X(4) + X(6)
X(14561) = 2 X(5) + X(6) = X(4) + 2 X(182) = X(69) + 2 X(576) = X(381) + 2 X(597) = 4 X(547) - X(599) = 4 X(140) - X(1350) = 2 X(141) + X(1351) = 4 X(5) - X(1352) = 2 X(6) + X(1352) = 5 X(6) - 2 X(1353) = 5 X(5) + X(1353) = 5 X(1352) + 4 X(1353) = X(355) + 2 X(1386) = 2 X(141) - 5 X(1656) = X(1351) + 5 X(1656) = X(69) - 7 X(3090) = 2 X(576) + 7 X(3090) = 4 X(575) + 5 X(3091) = 5 X(631) - 2 X(3098) = X(3) - 4 X(3589) = 2 X(182) - 5 X(3618) = X(4) + 5 X(3618) = 8 X(3628) - 5 X(3763) = 5 X(3091) - 2 X(3818) = 2 X(575) + X(3818) = X(193) + 11 X(5056) = 7 X(3619) - 13 X(5067) = X(1992) + 5 X(5071) = 2 X(3629) + 13 X(5079) = X(20) - 4 X(5092) = 3 X(5055) + X(5093) = X(193) - 4 X(5097) = 11 X(5056) + 4 X(5097) = 4 X(547) + X(5102) = X(3313) + 2 X(5446) = X(2) + 2 X(5476) = 2 X(3589) + X(5480) = X(3) + 2 X(5480) = X(3242) - 4 X(5901) = 13 X(5068) - X(5921) = 2 X(5026) + X(6321) = 7 X(3851) + 8 X(6329) = X(265) + 2 X(6593) = 4 X(575) - X(6776) = 5 X(3091) + X(6776) = 2 X(3818) + X(6776) = 5 X(3620) - 17 X(7486) = 4 X(7606) - X(8182) = X(3751) + 5 X(8227) = 4 X(6329) - X(8550) = 7 X(3851) + 2 X(8550) = 4 X(206) - X(9833) = X(3416) - 4 X(9956) = X(9967) + 2 X(9969) = 2 X(125) + X(9970) = X(376) - 4 X(10168) = X(2930) - 4 X(10272) = 2 X(1353) + 5 X(10516) = 6 X(5476) + X(10519) = 2 X(3627) + 7 X(10541) = X(7737) - 4 X(10796) = 5 X(5071) - 2 X(11178) = X(1992) + 2 X(11178) = 4 X(597) - X(11179) = 2 X(381) + X(11179) = X(3094) - 4 X(11272) = 8 X(3628) + X(11477) = 5 X(3763) + X(11477) = 2 X(3629) - 5 X(11482) = 13 X(5079) + 5 X(11482) = 2 X(10110) + X(11574) = 2 X(113) + X(11579) = 4 X(1843) - X(11663) = 13 X(5079) - X(11898) = 2 X(3629) + X(11898) = 5 X(11482) + X(11898) = 11 X(5072) + 4 X(12007) = X(382) + 5 X(12017) = 2 X(115) + X(12177) = X(5181) - 4 X(12900) = X(9971) - 4 X(13364) = 2 X(2030) + X(13449) = 4 X(8177) - X(14023) = X(10753) + 5 X(14061) = X(5967) + 2 X(14356)

X(14561) lies on these lines: {2, 51}, {3, 3589}, {4, 83}, {5, 6}, {11, 611}, {12, 613}, {20, 5092}, {25, 13394}, {30, 5085}, {66, 5576}, {69, 576}, {113, 11579}, {114, 7736}, {115, 5034}, {125, 9970}, {140, 1350}, {141, 1351}, {159, 7529}, {184, 6997}, {193, 5056}, {206, 569}, {265, 6593}, {343, 7539}, {355, 1386}, {376, 10168}, {381, 597}, {382, 12017}, {389, 7404}, {427, 10601}, {428, 3796}, {458, 6530}, {468, 3066}, {498, 3056}, {499, 1469}, {518, 5886}, {524, 5055}, {542, 3545}, {547, 599}, {575, 3091}, {578, 7401}, {631, 3098}, {732, 7697}, {1428, 1478}, {1479, 2330}, {1506, 5028}, {1513, 11174}, {1570, 7603}, {1598, 3867}, {1691, 7737}, {1692, 5475}, {1843, 3542}, {1899, 5133}, {1992, 5071}, {1995, 14389}, {2030, 13449}, {2065, 5967}, {2782, 6034}, {2930, 10272}, {3054, 11173}, {3088, 9729}, {3094, 11272}, {3095, 7795}, {3242, 5901}, {3311, 13910}, {3312, 13972}, {3313, 5446}, {3329, 9744}, {3416, 9956}, {3541, 12294}, {3546, 11695}, {3547, 10110}, {3549, 9967}, {3574, 6816}, {3619, 5067}, {3620, 7486}, {3627, 10541}, {3628, 3763}, {3629, 5079}, {3751, 8227}, {3832, 11572}, {3839, 11645}, {3851, 6329}, {4259, 6862}, {4260, 6824}, {4265, 6924}, {5012, 7394}, {5026, 6321}, {5033, 7747}, {5039, 7735}, {5052, 7746}, {5068, 5921}, {5072, 12007}, {5096, 6914}, {5135, 6917}, {5138, 6826}, {5181, 12900}, {5286, 6248}, {5790, 5846}, {5800, 6893}, {5847, 10175}, {5999, 7875}, {6144, 12812}, {6230, 13640}, {6231, 13760}, {6403, 7505}, {6721, 8781}, {6803, 13346}, {6815, 11424}, {6832, 10477}, {7392, 9306}, {7399, 10982}, {7400, 13598}, {7403, 14216}, {7533, 11003}, {7544, 13434}, {7558, 9781}, {7606, 8182}, {7693, 14002}, {7741, 12589}, {7789, 10983}, {7791, 12110}, {7792, 13860}, {7808, 13355}, {7834, 13354}, {7951, 12588}, {8177, 14023}, {9053, 10247}, {9737, 14001}, {9738, 11291}, {9739, 11292}, {9825, 11425}, {9862, 10348}, {9971, 10201}, {10104, 12212}, {10753, 14061}, {11064, 11284}, {11185, 12215}, {11479, 12233}, {13692, 13769}, {13812, 13833}

X(14561) = midpoint of X(i) and X(j) for these {i,j}: {6, 10516}, {381, 5050}, {599, 5102}
X(14561) = reflection of X(i) in X(j) for these {i,j}: {1352, 10516}, {5050, 597}, {10516, 5}, {11179, 5050}
X(14561) = complement X(10519)
X(14561) = centroid of the triangle X(3)X(4)X(6)}
X(14561) = crossdifference of every pair of points on line {924, 3288}
X(14561) = X(2)-of-X(6)PU(5)
X(14561) = X(3)X(4)X(6)-isogonal conjugate of X(33753)
X(14561) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4, 3618, 182), (5, 6, 1352), (5, 7583, 6290), (5, 7584, 6289), (381, 597, 11179), (485, 486, 2548), (569, 7528, 9833), (575, 3818, 6776), (1351, 1656, 141), (1992, 5071, 11178), (2546, 2547, 10359), (3091, 6776, 3818), (3329, 13862, 9744), (3589, 5480, 3), (5133, 5422, 1899), (7392, 11427, 9306), (7539, 9777, 343), (11482, 11898, 3629)

X(14562) =  (name pending)

Barycentrics    (a - 2*b - 2*c)*(a + b - c)*(a - b + c)*(4*a^4 - 7*a^3*b - 3*a^2*b^2 + 7*a*b^3 - b^4 - 7*a^3*c - 2*a^2*b*c - 7*a*b^2*c - 3*a^2*c^2 - 7*a*b*c^2 + 2*b^2*c^2 + 7*a*c^3 - c^4) : :

X(14562) lies on this line: {65, 513}

X(14563) =  MIDPOINT OF X(1) AND X(11041)

Barycentrics    4*a^4 - 7*a^3*b - 3*a^2*b^2 + 7*a*b^3 - b^4 - 7*a^3*c - 2*a^2*b*c - 7*a*b^2*c - 3*a^2*c^2 - 7*a*b*c^2 + 2*b^2*c^2 + 7*a*c^3 - c^4 : :
X(14563) = 3 X(1) - X(1000) = 5 X(1000) - 3 X(8275) = 5 X(1) - X(8275) = X(1000) + 3 X(11041) = X(8275) + 5 X(11041) = X(7) - 3 X(11529)

X(14563) lies on the cubic K573 and these lines: {1, 631}, {7, 515}, {10, 5719}, {65, 4304}, {80, 226}, {142, 519}, {145, 11024}, {354, 1317}, {355, 12563}, {514, 4667}, {516, 1159}, {517, 5572}, {938, 13464}, {942, 4315}, {946, 5722}, {952, 5542}, {1387, 9952}, {1482, 6744}, {2800, 5728}, {3241, 3306}, {3244, 5045}, {3333, 7966}, {3487, 5726}, {3625, 9710}, {3635, 3913}, {3671, 11544}, {3754, 12437}, {4029, 4752}, {4031, 5902}, {4084, 12917}, {4301, 12433}, {4867, 5316}, {5219, 10175}, {5795, 12559}, {5881, 11036}, {7967, 10980}, {10572, 11552}, {11023, 11518}

X(14563) = midpoint of X(i) and X(j) for these {i,j}: {1, 11041}, {145, 11525}
X(14563) = reflection of X(7966) in X(13607)
X(14563) = complement of X(36922)

X(14564) =  X(7)X(44)∩X(65)X(513)

Barycentrics    (a + b - c)*(a - b + c)*(4*a^3 - 5*a^2*b - a*b^2 + 2*b^3 - 5*a^2*c + 2*a*b*c - 2*b^2*c - a*c^2 - 2*b*c^2 + 2*c^3) : :

X(14564) lies on these lines: on lines {1, 11662}, {7, 44}, {65, 513}, {85, 4715}, {279, 4644}, {348, 4795}, {527, 1212}, {3668, 7277}, {4419, 5543}, {4675, 12848}

X(14565) =  X(323)X(2502)∩X(543)X(7799)

Barycentrics    a^2*(a^6 - a^4*b^2 - a^2*b^4 + b^6 - 2*a^4*c^2 + a^2*b^2*c^2 - 2*b^4*c^2 + 3*a^2*c^4 + 3*b^2*c^4 - 3*c^6)*(a^6 - 2*a^4*b^2 + 3*a^2*b^4 - 3*b^6 - a^4*c^2 + a^2*b^2*c^2 + 3*b^4*c^2 - a^2*c^4 - 2*b^2*c^4 + c^6) : :

X(14565) lies on these lines: {323, 2502}, {543, 7799}, {575, 13137}

X(14565) = trilinear pole of line {526, 9171}

X(14566) =  X(2)X(525)∩X(5)X(523)

Barycentrics    (b - c)*(b + c)*(a^8 - 4*a^6*b^2 + 6*a^4*b^4 - 4*a^2*b^6 + b^8 - 4*a^6*c^2 + a^4*b^2*c^2 + a^2*b^4*c^2 + 2*b^6*c^2 + 6*a^4*c^4 + a^2*b^2*c^4 - 6*b^4*c^4 - 4*a^2*c^6 + 2*b^2*c^6 + c^8) : :
X(14566) = 3 X(2) + X(2394) = 5 X(1656) + X(5489) = 5 X(14061) - X(14223)

X(14566) lies on these lines: {2, 525}, {5, 523}, {520, 10170}, {690, 6036}, {1499, 9756}, {1637, 6334}, {1649, 11637}, {1656, 5489}, {2799, 5461}, {6704, 8574}, {14061, 14223}

X(14566) = midpoint of X(i) and X(j) for these {i,j}: {1637, 6334}, {2394, 5664}
X(14566) = complement X(5664)
X(14566) = X(2159)-complementary conjugate of X(3258)
X(14566) = X(3268)-Ceva conjugate of X(523)
X(14566) = X(163)-isoconjugate of X(1138)
X(14566) = crossdifference of every pair of points on line {50, 1495}
X(14566) = {X(2),X(2394)}-harmonic conjugate of X(5664)
X(14566) = PU(5)-harmonic conjugate of X(18121)
X(14566) = barycentric product X(i)*X(j) for these {i,j}: {399, 850}, {523, 1272}
X(14566) = barycentric quotient X(i)/X(j) for these {i,j}: {399, 110}, {523, 1138}, {1272, 99}, {1637, 11070}, {8562, 14354}

X(14567) =  CROSSSUM OF X(2) AND X(316)

Barycentrics    a^4*(2*a^2 - b^2 - c^2) : :

X(14567) lies on these lines: {6, 23}, {32, 184}, {50, 3289}, {110, 1691}, {187, 3292}, {251, 14153}, {323, 2076}, {394, 5023}, {468, 1648}, {511, 8627}, {575, 13410}, {669, 688}, {698, 10330}, {1495, 1692}, {1613, 9544}, {1627, 2056}, {1915, 5012}, {2021, 5191}, {2030, 2502}, {3266, 5026}, {3796, 5013}, {5033, 5651}, {5041, 11205}, {6660, 9605}, {9604, 13338}, {11422, 13330}

X(14567) = isogonal conjugate of X(18023)
X(14567) = crosspoint of X(6) and X(3455)
X(14567) = crossdifference of every pair of points on line {76, 850}
X(14567) = crosssum of X(i) and X(j) for these (i,j): {2, 316}, {76, 3266}, {115, 9134}
X(14567) = X(2491)-Hirst inverse of X(9407)
X(14567) = polar conjugate of isotomic conjugate of X(23200)
X(14567) = barycentric product of PU(107)
X(14567) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (110, 1691, 3231), (184, 1501, 3051), (1495, 1692, 3124)
X(14567) = X(i)-isoconjugate of X(j) for these (i,j): {75, 671}, {76, 897}, {92,207a86}, {111, 561}, {799, 5466}, {892, 1577}, {895, 1969}, {923, 1502}, {3261, 5380}, {4602, 9178}
X(14567) = barycentric product X(i)*X(j) for these {i,j}: {1, 922}, {6, 187}, {25, 3292}, {31, 896}, {32, 524}, {110, 351}, {163, 2642}, {184, 468}, {237, 5967}, {512, 5467}, {560, 14210}, {669, 5468}, {690, 1576}, {692, 14419}, {1397, 3712}, {1495, 9717}, {1501, 3266}, {1918, 6629}, {1922, 4760}, {1974, 6390}, {1976, 9155}, {2175, 7181}, {2206, 4062}, {2434, 8644}, {3049, 4235}, {3455, 6593}, {4630, 14424}, {5026, 9468}, {7104, 7267}, {14270, 14559}
X(14567) = barycentric quotient X(i)/X(j) for these {i,j}: {32, 671}, {187, 76}, {351, 850}, {524, 1502}, {560, 897}, {669, 5466}, {896, 561}, {922, 75}, {1501, 111}, {1576, 892}, {1917, 923}, {3292, 305}, {5467, 670}, {5468, 4609}, {9407, 9214}, {9418, 5968}, {9426, 9178}, {9448, 5547}, {14210, 1928}

X(14568) =  X(2)X(39)∩X(30)X(98)

Barycentrics   a^4 + b^4 - 3*b^2*c^2 + c^4 : :
X(14568) = X(148) + 2 X(187) = X(99) - 4 X(230) = 4 X(115) - X(316) = 2 X(115) + X(385) = X(316) + 2 X(385) = 4 X(625) - X(7779) = 4 X(6722) - X(7813) = 4 X(5461) - X(7840) = 8 X(6722) - 5 X(7925) = 2 X(7813) - 5 X(7925) = X(7809) - 3 X(9166) = 4 X(3934) - X(9865) = 2 X(2) + X(11054) = X(5184) + 2 X(11599) = X(5999) - 4 X(11623) = 3 X(8859) - X(13586) = 2 X(325) - 5 X(14061)

Let (Oa) be the circle whose diameter is the segment cut off by the orthogonal projections of PU(2) on line BC. Define (Ob) and (Oc) cyclically. X(14568) is the radical center of (Oa), (Ob), (Oc). (Randy Hutson, November 2, 2017)

X(14568) lies on the cubic K492 and these lines: {2, 39}, {4, 6179}, {5, 7760}, {30, 98}, {32, 11361}, {69, 7934}, {83, 5305}, {99, 230}, {115, 316}, {141, 7919}, {148, 187}, {183, 7790}, {193, 7926}, {325, 14061}, {381, 7812}, {384, 7755}, {403, 648}, {491, 13711}, {492, 13834}, {524, 5103}, {532, 5460}, {533, 5459}, {543, 5152}, {597, 12151}, {598, 7615}, {625, 7779}, {626, 14046}, {637, 13886}, {638, 13939}, {1078, 5254}, {1352, 1992}, {1506, 7839}, {1975, 7857}, {1989, 3228}, {2021, 9890}, {2039, 6189}, {2040, 6190}, {2367, 9091}, {2489, 4580}, {2548, 7894}, {2549, 7771}, {2896, 7861}, {3096, 7851}, {3314, 7844}, {3329, 5355}, {3734, 7806}, {3785, 7910}, {3830, 9873}, {3933, 7899}, {3972, 7735}, {4664, 10197}, {5025, 7751}, {5054, 8860}, {5055, 11163}, {5184, 11599}, {5306, 8370}, {5319, 7878}, {5346, 7787}, {5461, 7840}, {5475, 7766}, {5999, 11623}, {6292, 7923}, {6655, 7780}, {6722, 7813}, {6785, 13754}, {7664, 9870}, {7745, 12156}, {7748, 7793}, {7749, 7783}, {7752, 7754}, {7758, 7814}, {7765, 7824}, {7767, 7911}, {7770, 7856}, {7773, 7877}, {7775, 7837}, {7777, 7798}, {7781, 7907}, {7785, 7805}, {7788, 11318}, {7794, 7901}, {7796, 7887}, {7800, 7918}, {7808, 7920}, {7810, 7924}, {7811, 7841}, {7815, 7864}, {7825, 7893}, {7826, 7885}, {7833, 11648}, {7854, 7933}, {7855, 7912}, {7860, 14023}, {7862, 7906}, {7869, 14065}, {7872, 7904}, {7876, 7902}, {7890, 7941}, {7909, 8361}, {7922, 14064}, {8364, 10159}, {10000, 11286}

X(14568) = reflection of X(i) in X(j) for these {i,j}: {316, 14041}, {7799, 2}, {12151, 597}, {14041, 115}
X(14568) = X(2491)-Hirst inverse of X(9407)
midpoint of X(i) and X(j) for these {i,j}: {385, 14041}, {7799, 11054}
X(14568) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 5309, 7827), (5, 7760, 7858), (76, 3767, 7828), (76, 7828, 7832), (76, 7942, 7795), (115, 385, 316), (183, 7790, 7831), (194, 7746, 7769), (1078, 5254, 7847), (3934, 7797, 7859), (5025, 7751, 7768), (5254, 13468, 8356), (5306, 8370, 12150), (6722, 7813, 7925), (7735, 11185, 3972), (7752, 7754, 7905), (7754, 13881, 7752), (7817, 9466, 2), (7841, 8667, 7811), (8356, 13468, 1078), (14023, 14063, 7860)
X(14568) = X(798)-isoconjugate of X(2858)
X(14568) = barycentric product X(i)*X(j) for these {i,j}: {670, 2872}, {2510, 6331}
X(14568) = barycentric quotient X(i)/X(j) for these {i,j}: {99, 2858}, {2510, 647}, {2872, 512}

X(14569) =  X(5)X(324)∩X(51)X(53)

Barycentrics   (a^2 + b^2 - c^2)^2*(a^2 - b^2 + c^2)^2*(a^2*b^2 - b^4 + a^2*c^2 + 2*b^2*c^2 - c^4) : :

X(14569) lies on these lines: {2, 14489}, {4, 3527}, {5, 324}, {25, 393}, {51, 53}, {107, 5966}, {184, 1990}, {235, 1093}, {262, 427}, {297, 12251}, {428, 14495}, {460, 2207}, {462, 3490}, {463, 3489}, {468, 11547}, {1249, 11402}, {1629, 10301}, {1896, 1904}, {1906, 14249}, {5064, 10002}, {6747, 13567}, {6750, 14363}, {6995, 9755}, {8794, 8901}

X(14569) = X(i)-Ceva conjugate of X(j) for these (i,j): {393, 3199}, {13450, 53}
X(14569) = X(3199)-cross conjugate of X(53)
X(14569) = X(i)-isoconjugate of X(j) for these (i,j): {54, 326}, {63, 97}, {69, 2169}, {95, 255}, {275, 6507}, {276, 4100}, {304, 14533}, {394, 2167}, {1102, 8882}, {2148, 3926}, {2190, 3964}
X(14569) = crosspoint of X(393) and X(1093)
X(14569) = crosssum of X(394) and X(1092)
X(14569) = polar conjugate of X(34386)
X(14569) = perspector of ABC and orthoanticevian triangle of X(53)
X(14569) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (51, 53, 6755), (393, 6524, 25), (2052, 6530, 427)
X(14569) = barycentric product X(i)*X(j) for these {i,j}: {4, 53}, {5, 393}, {6, 13450}, {25, 324}, {51, 2052}, {92, 2181}, {107, 12077}, {158, 1953}, {216, 1093}, {264, 3199}, {311, 2207}, {343, 6524}, {1096, 14213}, {6116, 8738}, {6117, 8737}, {6344, 11062}, {6368, 6529}, {6525, 13157}, {6755, 8796}
X(14569) = barycentric quotient X(i)/X(j) for these {i,j}: {5, 3926}, {25, 97}, {51, 394}, {53, 69}, {216, 3964}, {217, 1092}, {324, 305}, {343, 4176}, {393, 95}, {1093, 276}, {1096, 2167}, {1393, 7183}, {1953, 326}, {1973, 2169}, {1974, 14533}, {2179, 255}, {2181, 63}, {2207, 54}, {3199, 3}, {6368, 4143}, {6524, 275}, {7069, 3719}, {12077, 3265}, {13450, 76}

X(14570) =  ISOGONAL CONJUGATE OF X(2623)

Barycentrics    cot(B - C) : :
Barycentrics    (a - b)*(a + b)*(a - c)*(a + c)*(a^2*b^2 - b^4 + a^2*c^2 + 2*b^2*c^2 - c^4) : :

X(14570) lies on these lines: {2, 94}, {3, 9512}, {20, 2781}, {69, 1972}, {98, 14060}, {99, 112}, {110, 925}, {114, 8754}, {194, 1992}, {216, 311}, {523, 1634}, {645, 2397}, {655, 662}, {670, 2396}, {895, 11596}, {1576, 4226}, {2421, 4576}, {2452, 13188}, {2966, 4577}, {5095, 10992}, {8724, 9214}, {9308, 9723}

X(14570) = isogonal conjugate of X(2623)
X(14570) = isotomic conjugate of X(15412)
X(14570) = anticomplement X(338)
X(14570) = X(249)-Ceva conjugate of X(2)
X(14570) = X(i)-cross conjugate of X(j) for these (i,j): {6368, 311}, {12077, 5}
X(14570) = polar conjugate of isogonal conjugate of X(23181)
X(14570) = cevapoint of X(i) and X(j) for these (i,j): {5, 12077}, {216, 6368}, {523, 570}, {647, 6146}, {3575, 6753}
X(14570) = crosspoint of X(99) and X(6331)
X(14570) = trilinear pole of line X(5)X(51)
X(14570) = crosssum of X(512) and X(3049)
X(14570) = X(99)-daleth conjugate of X(877)
X(14570) = X(99)-waw conjugate of X(4576)
X(14570) = Jerabek image of X(4)
X(14570) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {163, 3448}, {249, 6327}, {1101, 69}, {4575, 13219}
X(14570) = X(i)-isoconjugate of X(j) for these (i,j): {1, 2623}, {6, 2616}, {19,23286}, {54, 661}, {95, 798}, {163, 8901}, {275, 810}, {512, 2167}, {523, 2148}, {647, 2190}, {656, 8882}, {822, 8884}, {924, 2168}, {933, 3708}, {1141, 2624}, {2169, 2501}
X(14570) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (99, 648, 4558), (648, 4558, 2407), (4552, 4560, 662)
X(14570) = barycentric product X(i)*X(j) for these {i,j}: {5, 99}, {51, 670}, {53, 4563}, {75, 2617}, {76, 1625}, {110, 311}, {216, 6331}, {324, 4558}, {343, 648}, {476, 1273}, {662, 14213}, {799, 1953}, {1393, 7257}, {2179, 4602}, {4590, 12077}, {4625, 7069}, {5562, 6528}
X(14570) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 2616}, {5, 523}, {6, 2623}, {51, 512}, {52, 924}, {53, 2501}, {99, 95}, {107, 8884}, {110, 54}, {112, 8882}, {143, 1510}, {162, 2190}, {163, 2148}, {216, 647}, {217, 3049}, {250, 933}, {311, 850}, {343, 525}, {476, 1141}, {523, 8901}, {648, 275}, {662, 2167}, {925, 96}, {930, 252}, {1087, 2618}, {1154, 526}, {1273, 3268}, {1393, 4017}, {1568, 9033}, {1625, 6}, {1953, 661}, {2081, 2088}, {2179, 798}, {2290, 2624}, {2617, 1}, {2618, 1109}, {2621, 2627}, {2625, 2619}, {2626, 2620}, {3199, 2489}, {4558, 97}, {4575, 2169}, {5562, 520}, {5891, 8675}, {6331, 276}, {6368, 125}, {6528, 8795}, {7069, 4041}, {9387, 9389}, {12077, 115}, {14213, 1577}
X(14570) =

X(14571) =  CROSSPOINT OF X(2) AND X(1295)

Barycentrics    a*(a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*(a^2*b - b^3 + a^2*c - 2*a*b*c + b^2*c + b*c^2 - c^3) : :
Trilinears    (tan A) (cos B + cos C - 1) : :

X(14571) lies on these lines: {4, 4277}, {6, 19}, {37, 158}, {42, 1859}, {44, 1783}, {45, 7079}, {53, 1826}, {55, 1096}, {92, 3666}, {112, 2687}, {196, 1427}, {197, 7337}, {204, 3052}, {216, 828}, {227, 1118}, {230, 231}, {240, 518}, {241, 653}, {243, 9371}, {278, 3752}, {386, 1871}, {910, 7115}, {1013, 4689}, {1086, 5236}, {1108, 1148}, {1155, 1430}, {1172, 1389}, {1407, 1767}, {1486, 6059}, {1748, 4641}, {1824, 3192}, {1839, 6748}, {1865, 2092}, {1870, 9456}, {1875, 2183}, {1888, 4642}, {1957, 4640}, {2178, 3209}, {2271, 3553}, {2294, 2658}, {3194, 6197}, {3330, 6001}, {3445, 7129}, {3683, 7076}, {5336, 8573}, {7719, 8557}

X(14571) = X(1295)-complementary conjugate of X(2887)
X(14571) = X(i)-Ceva conjugate of X(j) for these (i,j): {278, 1846}, {915, 25}, {2405, 6087}
X(14571) = X(i)-isoconjugate of X(j) for these (i,j): {2, 1795}, {57, 1809}, {63, 104}, {69, 909}, {348, 2342}, {1309, 4091}, {1331, 2401}, {1444, 2250}, {1459, 13136}, {2423, 4561}, {2720, 6332}, {3977, 10428}
X(14571) = crosspoint of X(2) and X(1295)
X(14571) = crossdifference of every pair of points on line {3, 521}
X(14571) = crosssum of X(i) and X(j) for these (i,j): {6, 6001}, {219, 5440}
X(14571) = X(1295)-complementary conjugate of X(2887)
X(14571) = pole wrt polar circle of trilinear polar of X(18816) (line X(2)X(905))
X(14571) = polar conjugate of X(18816)
X(14571) = X(63)-isoconjugate of X(104)
X(14571) = crosspoint of X(4) and X(16082)
X(14571) = X(i)-isoconjugate of X(j) for these (i,j): {1, 2623}, {6, 2616}, {54, 661}, {95, 798}, {163, 8901}, {275, 810}, {512, 2167}, {523, 2148}, {647, 2190}, {656, 8882}, {822, 8884}, {924, 2168}, {933, 3708}, {1141, 2624}, {2169, 2501}
X(14571) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (19, 1880, 1841), (19, 2331, 6), (42, 2181, 1859), (1886, 8755, 1990), (1886, 8756, 8755)
X(14571) = barycentric product X(i)*X(j) for these {i,j}: {1, 1785}, {4, 517}, {8, 1875}, {19, 908}, {25, 3262}, {34, 6735}, {80, 1845}, {92, 2183}, {108, 2804}, {119, 915}, {281, 1465}, {318, 1457}, {523, 4246}, {1320, 1846}, {1769, 1897}, {1783, 10015}, {2397, 6591}, {3310, 6335}, {8072, 8073}
X(14571) = barycentric quotient X(i)/X(j) for these {i,j}: {25, 104}, {31, 1795}, {55, 1809}, {517, 69}, {859, 1444}, {908, 304}, {1457, 77}, {1465, 348}, {1769, 4025}, {1783, 13136}, {1785, 75}, {1845, 320}, {1875, 7}, {1973, 909}, {2183, 63}, {2212, 2342}, {2333, 2250}, {2427, 1332}, {3262, 305}, {3310, 905}, {4246, 99}, {6591, 2401}, {6735, 3718}, {8677, 4131}

X(14572) = X(3146)-CROSS CONJUGATE OF X(253)

Barycentrics   (a^4 - 2*a^2*b^2 + b^4 + 2*a^2*c^2 + 2*b^2*c^2 - 3*c^4)*(5*a^4 - 2*a^2*b^2 - 3*b^4 - 2*a^2*c^2 + 6*b^2*c^2 - 3*c^4)*(a^4 + 2*a^2*b^2 - 3*b^4 - 2*a^2*c^2 + 2*b^2*c^2 + c^4) : :

X(14572) lies on the cubic K347 and these lines: {2, 253}, {20, 6526}, {64, 3091}, {1968, 11348}, {5922, 10192}, {10303, 14379}

X(14572) = X(3146)-cross conjugate of X(253)
X(14572) = X(610)-isoconjugate of X(3532)
X(14572) = {X(2),X(459)}-harmonic conjugate of X(253)
X(14572) = barycentric product X(253)*X(3146)
X(14572) = barycentric quotient X(i)/X(j) for these {i,j}: {64, 3532}, {3146, 20}, {13611, 122}
X(14572) = polar conjugate of X(33893)
X(14572) = polar conjugate of Kirikami-Euler image of X(20)

X(14573) =  X(6)X(2934)∩X(54)X(511)

Barycentrics   a^6*(a^4 - 2*a^2*b^2 + b^4 - a^2*c^2 - b^2*c^2)*(a^4 - a^2*b^2 - 2*a^2*c^2 - b^2*c^2 + c^4) : :

X(14573) lies on these lines: {6, 2934}, {54, 511}, {160, 184}, {1843, 1976}, {5562, 8883}

X(14573) = X(i)-isoconjugate of X(j) for these (i,j): {5, 561}, {51, 1928}, {75, 311}, {76, 14213}, {304, 324}, {343, 1969}, {670, 2618}, {1502, 1953}, {4602, 12077}
X(14573) = barycentric product X(i)*X(j) for these {i,j}: {25, 14533}, {31, 2148}, {32, 54}, {95, 1501}, {97, 1974}, {184, 8882}, {560, 2167}, {933, 3049}, {1576, 2623}, {1973, 2169}, {2190, 9247}
X(14573) = barycentric quotient X(i)/X(j) for these {i,j}: {32, 311}, {54, 1502}, {560, 14213}, {1501, 5}, {1917, 1953}, {1924, 2618}, {1974, 324}, {2148, 561}, {2167, 1928}, {9233, 51}, {9426, 12077}, {14533, 305}

X(14574) =  BARYCENTRIC PRODUCT X(32)*X(110)

Barycentrics   a^6*(a - b)*(a + b)*(a - c)*(a + c) : :

X(14574) lies on these lines: {32, 2909}, {110, 827}, {112, 512}, {1501, 9427}, {1576, 14270}, {2387, 10317}, {3202, 9419}, {3203, 10547}, {11557, 12228}

X(14574) = X(4630)-Ceva conjugate of X(1576)
X(14574) = X(9426)-cross conjugate of X(1501)
X(14574) = cevapoint of X(i) and X(j) for these (i,j): {1501, 9426}, {3049, 3051}
X(14574) = trilinear pole of line X(1501)X(9233)
X(14574) = crosssum of X(i) and X(j) for these (i,j): {338, 8029}, {850, 3267}
X(14574) = X(i)-isoconjugate of X(j) for these (i,j): {75, 850}, {76, 1577}, {92, 3267}, {115, 4602}, {264, 14208}, {310, 4036}, {313, 693}, {321, 3261}, {338, 799}, {339, 811}, {349, 4391}, {512, 1928}, {523, 561}, {525, 1969}, {661, 1502}, {670, 1109}, {871, 4122}, {1934, 14295}, {2643, 4609}, {3120, 6386}, {3596, 4077}, {3801, 7034}, {4024, 6385}, {4086, 6063}, {4143, 6521}
X(14574) = barycentric product X(i)*X(j) for these {i,j}: {6, 1576}, {31, 163}, {32, 110}, {39, 4630}, {50, 14560}, {99, 1501}, {101, 2206}, {112, 184}, {162, 9247}, {217, 933}, {237, 2715}, {249, 669}, {250, 3049}, {560, 662}, {670, 9233}, {692, 1333}, {798, 1101}, {799, 1917}, {827, 3051}, {906, 2203}, {1397, 5546}, {1414, 9447}, {1415, 2194}, {1918, 4556}, {1919, 4570}, {1923, 4599}, {1973, 4575}, {1974, 4558}, {1980, 4567}, {2175, 4565}, {2966, 9418}, {4573, 9448}, {4590, 9426}, {4591, 9459}
X(14574) = barycentric quotient X(i)/X(j) for these {i,j}: {32, 850}, {110, 1502}, {163, 561}, {184, 3267}, {249, 4609}, {560, 1577}, {662, 1928}, {669, 338}, {1101, 4602}, {1501, 523}, {1576, 76}, {1917, 661}, {1924, 1109}, {2205, 4036}, {2206, 3261}, {3049, 339}, {4630, 308}, {9233, 512}, {9247, 14208}, {9418, 2799}, {9426, 115}, {9427, 8029}, {9447, 4086}, {9448, 3700}

X(14575) =  X(3)X(1176)∩X(6)X(157)

Barycentrics   a^6*(a^2 - b^2 - c^2) : :

X(14575) lies on these lines: {3, 1176}, {6, 157}, {25, 8745}, {31, 2908}, {32, 682}, {50, 160}, {66, 14003}, {184, 418}, {206, 237}, {250, 2452}, {264, 9512}, {560, 9448}, {827, 3972}, {1501, 9233}, {2967, 3425}, {3167, 3964}, {3284, 6467}, {3398, 3618}, {5157, 14096}

X(14575) = isogonal conjugate of X(18022)
X(14575) = X(i)-Ceva conjugate of X(j) for these (i,j): {32, 1501}, {1576, 3049}, {10547, 184}
X(14575) = crosspoint of X(i) and X(j) for these (i,j): {6, 1485}, {31, 7139}, {32, 184}
X(14575) = crossdifference of every pair of points on line {2799, 3267}
X(14575) = X(14575) = crosssum of X(i) and X(j) for these (i,j): {2, 11442}, {69, 7763}, {76, 264}
X(14575) = X(i)-isoconjugate of X(j) for these (i,j): {2, 1969}, {4, 561}, {19, 1502}, {25, 1928}, {75, 264}, {76, 92}, {85, 7017}, {158, 305}, {273, 3596}, {276, 14213}, {286, 313}, {304, 2052}, {312, 331}, {318, 6063}, {811, 850}, {823, 3267}, {873, 7141}, {1235, 3112}, {1577, 6331}, {1826, 6385}, {2501, 4602}, {2973, 7035}, {3261, 6335}, {3926, 6521}, {6386, 7649}, {6528, 14208}
X(14575) = X(1501)-Hirst inverse of X(9418)
X(14575) = {X(206),X(571)}-harmonic conjugate of X(237)
X(14575) = barycentric product X(i)*X(j) for these {i,j}: {1, 9247}, {3, 32}, {6, 184}, {25, 577}, {31, 48}, {39, 10547}, {41, 603}, {51, 14533}, {54, 217}, {58, 2200}, {63, 560}, {69, 1501}, {71, 2206}, {77, 9447}, {110, 3049}, {163, 810}, {212, 604}, {213, 1437}, {219, 1397}, {222, 2175}, {228, 1333}, {237, 248}, {255, 1973}, {287, 9418}, {293, 9417}, {304, 1917}, {305, 9233}, {348, 9448}, {394, 1974}, {418, 8882}, {571, 2351}, {607, 7335}, {608, 6056}, {647, 1576}, {667, 906}, {669, 4558}, {798, 4575}, {1092, 2207}, {1096, 4100}, {1106, 1802}, {1176, 3051}, {1253, 7099}, {1331, 1919}, {1332, 1980}, {1395, 2289}, {1402, 2193}, {1409, 2194}, {1415, 1946}, {1444, 2205}, {1474, 4055}, {1790, 1918}, {1797, 9459}, {1814, 9455}, {1922, 7193}, {1924, 4592}, {1976, 3289}, {2169, 2179}, {2188, 2199}, {2196, 2210}, {2203, 3990}, {2212, 7125}, {2353, 10316}, {3172, 14379}, {3432, 8565}, {3455, 10317}, {3955, 7104}, {4563, 9426}, {5062, 8825}, {7114, 7118}, {7116, 7122}, {8577, 8911}, {8789, 12215}
X(14575) = barycentric quotient X(i)/X(j) for these {i,j}: {3, 1502}, {31, 1969}, {32, 264}, {48, 561}, {63, 1928}, {184, 76}, {217, 311}, {560, 92}, {577, 305}, {906, 6386}, {1084, 2970}, {1397, 331}, {1437, 6385}, {1501, 4}, {1576, 6331}, {1917, 19}, {1974, 2052}, {1977, 2973}, {2175, 7017}, {2200, 313}, {3049, 850}, {3051, 1235}, {4558, 4609}, {4575, 4602}, {7109, 7141}, {9233, 25}, {9247, 75}, {9418, 297}, {9426, 2501}, {9427, 8754}, {9447, 318}, {9448, 281}, {10547, 308}

X(14576) =  X(5)X(53)∩X(6)X(25)

Barycentrics   a^2*(a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*(a^2*b^2 - b^4 + a^2*c^2 + 2*b^2*c^2 - c^4)*(a^4 - 2*a^2*b^2 + b^4 - 2*a^2*c^2 + c^4) : :

X(14576) lies on the cubic K627 and thesse lines: {4, 570}, {5, 53}, {6, 25}, {24, 571}, {26, 577}, {32, 8746}, {39, 6748}, {112, 2383}, {393, 847}, {403, 1879}, {566, 7507}, {800, 1990}, {1166, 3518}, {1609, 2207}, {1968, 8553}, {1987, 8612}, {2965, 3432}, {3087, 5421}, {5158, 13861}, {6749, 7715}, {7514, 10979}, {8743, 13345}, {10282, 14533}

X(14576) = X(i)-Ceva conjugate of X(j) for these (i,j): {393, 53}, {467, 52}, {13450, 51}
X(14576) = X(i)-isoconjugate of X(j) for these (i,j): {63, 96}, {68, 2167}, {69, 2168}, {91, 97}, {95, 1820}, {2169, 5392}
X(14576) = crosspoint of X(i) and X(j) for these (i,j): {24, 11547}, {393, 8745}
X(14576) = crosssum of X(6) and X(9833)
X(14576) = polar conjugate of X(34385)
X(14576) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (24, 8745, 571), (53, 11062, 216), (216, 3199, 53), (393, 3542, 2165)
X(14576) = X(14576) = barycentric product X(i)*X(j) for these {i,j}: {4, 52}, {5, 24}, {6, 467}, {51, 317}, {53, 1993}, {92, 2180}, {143, 14111}, {216, 11547}, {324, 571}, {343, 8745}, {578, 14149}, {847, 3133}, {1147, 13450}, {1748, 1953}, {3199, 7763}
X(14576) = barycentric quotient X(i)/X(j) for these {i,j}: {24, 95}, {25, 96}, {51, 68}, {52, 69}, {53, 5392}, {467, 76}, {571, 97}, {1973, 2168}, {2179, 1820}, {2180, 63}, {2181, 91}, {3133, 9723}, {3199, 2165}, {8745, 275}, {11547, 276}

X(14577) =  BARYCENTRIC PRODUCT X(4)*X(143)

Barycentrics   a^2*(a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*(a^2*b^2 - b^4 + a^2*c^2 + 2*b^2*c^2 - c^4)*(a^4 - 2*a^2*b^2 + b^4 - 2*a^2*c^2 - b^2*c^2 + c^4) : :

X(14577) lies on these lines: {4, 13351}, {5, 53}, {6, 1173}, {25, 8746}, {232, 428}, {393, 1989}, {571, 2493}, {2965, 3518}, {6749, 13341}, {10641, 11088}, {10642, 11083}

X(14577) = X(14129)-Ceva conjugate of X(143)
X(14577) = X(i)-isoconjugate of X(j) for these (i,j): {63, 252}, {97, 2962}, {2167, 3519}, {2169, 11140}
X(14577) = barycentric product X(i)*X(j) for these {i,j}: {4, 143}, {5, 3518}, {6, 14129}, {49, 13450}, {53, 1994}, {137, 250}, {324, 2965}, {3199, 7769}
X(14577) = barycentric quotient X(i)/X(j) for these {i,j}: {25, 252}, {51, 3519}, {53, 11140}, {137, 339}, {143, 69}, {2181, 2962}, {2965, 97}, {3199, 2963}, {3518, 95}, {14129, 76}

X(14578) =  BARYCENTRIC PRODUCT X(3)*X(104)

Barycentrics    a^3*(a^2 - b^2 - c^2)*(a^3 - a^2*b - a*b^2 + b^3 + 2*a*b*c - a*c^2 - b*c^2)*(a^3 - a*b^2 - a^2*c + 2*a*b*c - b^2*c - a*c^2 + c^3) : :

X(14578) lies on these lines: {6, 909}, {104, 112}, {212, 2638}, {219, 577}, {654, 2423}, {1576, 2194}, {1812, 4558}, {2192, 2342}

X(14578) = isogonal conjugate of polar conjugate of X(104)
X(14578) = isotomic conjugate of polar conjugate of X(34858)
X(14578) = crosspoint of X(1295) and X(2990)
X(14578) = trilinear pole of line X(184)X(1946)
X(14578) = crossdifference of every pair of points on line {119, 2804}
X(14578) = crosssum of X(6001) and X(8609)
X(14578) = X(1726)-zayin conjugate of X(2183)
X(14578) = X(i)-isoconjugate of X(j) for these (i,j): {2, 1785}, {4, 908}, {19, 3262}, {92, 517}, {264, 2183}, {278, 6735}, {312, 1875}, {318, 1465}, {653, 2804}, {1145, 6336}, {1457, 7017}, {1577, 4246}, {1769, 6335}, {1846, 4997}, {1897, 10015}, {2397, 7649}
X(14578) = barycentric product X(i)*X(j) for these {i,j}: {1, 1795}, {3, 104}, {19,3262}, {56, 1809}, {63, 909}, {77, 2342}, {521, 2720}, {906, 2401}, {1332, 2423}, {1790, 2250}, {5440, 10428}
X(14578) = barycentric quotient X(i)/X(j) for these {i,j}: {3, 3262}, {31, 1785}, {48, 908}, {104, 264}, {184, 517}, {212, 6735}, {906, 2397}, {909, 92}, {1397, 1875}, {1576, 4246}, {1795, 75}, {1809, 3596}, {1946, 2804}, {2342, 318}, {9247, 2183}

X(14579) = X(50)-CROSS CONJUGATE OF X(6)

Barycentrics   a^2*(a^6 - a^4*b^2 - a^2*b^4 + b^6 - 3*a^4*c^2 + a^2*b^2*c^2 - 3*b^4*c^2 + 3*a^2*c^4 + 3*b^2*c^4 - c^6)*(a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6 - a^4*c^2 + a^2*b^2*c^2 + 3*b^4*c^2 - a^2*c^4 - 3*b^2*c^4 + c^6) : :

X(14579) lies on the circumconic {A,B,C,X(2), X(6)} and these lines: {2, 13582}, {3, 15553}, {6, 3200}, {111, 1291}, {115, 2963}, {231, 1989}, {251, 2493}, {1263, 2079}, {2081, 2433}, {3457, 11136}, {3458, 11135}, {3471, 12106}, {6128, 9698}, {8749, 11062}

X(14579) = isogonal conjugate of X(37779)
X(14579) = X(11071)-Ceva conjugate of X(6)
X(14579) = X(50)-cross conjugate of X(6)
X(14579) = cevapoint of X(3124) and X(14270)
X(14579) = trilinear pole of line X(512)X(13366)
X(14579) = circumcircle-inverse of X(15553)
X(14579) = crossdifference of every pair of points on line {6140, 6592}
X(14579) = crosssum of X(6) and X(5898)
X(14579) = X(i)-isoconjugate of X(j) for these (i,j): {2, 1749}, {75, 11063}, {799, 6140}, {1157, 14213}, {2349, 10272}, {3470, 14206}
X(14579) = barycentric product X(i)*X(j) for these {i,j}: {6, 13582}, {54, 1263}, {74, 3471}, {323, 11071}, {523, 1291}, {3459, 14367}
X(14579) = barycentric quotient X(i)/X(j) for these {i,j}: {31, 1749}, {32, 11063}, {669, 6140}, {1263, 311}, {1291, 99}, {1495, 10272}, {3124, 10413}, {3471, 3260}, {11071, 94}, {13582, 76}, {14270, 8562}

X(14580) = X(111)-CEVA CONJUGATE OF X(25)

Barycentrics    a^2*(a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*(a^4*b^2 - b^6 + a^4*c^2 - 2*a^2*b^2*c^2 + b^4*c^2 + b^2*c^4 - c^6) : :

X(14580) lies on these lines: {2, 1235}, {4, 9465}, {23, 112}, {25, 32}, {39, 5094}, {111, 8744}, {187, 8428}, {230, 231}, {421, 6531}, {427, 1194}, {852, 9475}, {858, 1560}, {1180, 8889}, {1995, 8743}, {2211, 3124}, {5158, 11284}, {5354, 10312}, {6353, 8879}, {6403, 9463}

X(14580) = X(i)-Ceva conjugate of X(j) for these (i,j): {111, 25}, {5523, 2393}
X(14580) = X(25)-vertex conjugate of X(2485)
X(14580) = crosspoint of X(i) and X(j) for these (i,j): {25, 8791}, {393, 8753}
X(14580) = crossdifference of every pair of points on line {3, 3265}
X(14580) = crosssum of X(394) and X(6390)
X(14580) = X(i)-isoconjugate of X(j) for these (i,j): {63, 2373}, {304, 1177}
X(14580) = PU(4)-harmonic conjugate of X(2485)
X(14580) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (232, 3291, 468), (232, 6103, 3003), (1990, 2493, 232)
X(14580) = barycentric product X(i)*X(j) for these {i,j}: {4, 2393}, {6, 5523}, {25, 858}, {111, 1560}, {1236, 1974}, {5181, 8753}
X(14580) = barycentric quotient X(i)/X(j) for these {i,j}: {25, 2373}, {858, 305}, {1560, 3266}, {1974, 1177}, {2393, 69}, {5523, 76}

X(14581) =  BARYCENTRIC PRODUCT X(25)*X(30)

Barycentrics    a^2*(a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*(2*a^4 - a^2*b^2 - b^4 - a^2*c^2 + 2*b^2*c^2 - c^4) : :

X(14581) lies on these lines: {4, 5007}, {6, 1597}, {25, 32}, {30, 1990}, {39, 378}, {112, 186}, {115, 10151}, {216, 7514}, {217, 13366}, {235, 7755}, {264, 7804}, {297, 754}, {393, 7737}, {403, 6103}, {427, 7753}, {512, 1692}, {538, 648}, {574, 11410}, {800, 8745}, {1249, 2549}, {1495, 9408}, {1593, 7772}, {1596, 5306}, {1885, 7765}, {2548, 8889}, {2794, 6530}, {3331, 8779}, {3440, 8739}, {3441, 8740}, {3734, 9308}, {3767, 6623}, {5008, 10311}, {5041, 13596}, {5052, 8541}, {5158, 9818}, {5206, 8778}, {5899, 10317}, {7865, 11331}, {10316, 12083}

X(14581) = X(i)-Ceva conjugate of X(j) for these (i,j): {1990, 1495}, {8749, 25}, {11060, 3199}
X(14581) = X(9407)-cross conjugate of X(1495)
X(14581) = X(i)-isoconjugate of X(j) for these (i,j): {63, 1494}, {69, 2349}, {74, 304}, {305, 2159}, {799, 14380}, {2394, 4592}
X(14581) = crosspoint of X(25) and X(8749)
X(14581) = crossdifference of every pair of points on line {69, 3265}
X(14581) = crosssum of X(69) and X(11064)
X(14581) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (32, 2207, 3199), (112, 232, 187), (112, 8744, 232), (1968, 8743, 39), (2207, 3172, 32)
X(14581) = barycentric product X(i)*X(j) for these {i,j}: {4, 1495}, {6, 1990}, {19, 2173}, {25, 30}, {31, 1784}, {92, 9406}, {107, 9409}, {112, 1637}, {264, 9407}, {393, 3284}, {512, 4240}, {607, 6357}, {608, 7359}, {648, 14398}, {1636, 6529}, {1783, 14399}, {1973, 14206}, {1974, 3260}, {2207, 11064}, {2407, 2489}, {2420, 2501}, {3163, 8749}, {3457, 6110}, {3458, 6111}, {5642, 8753}, {6525, 11589}, {8750, 11125}
X(14581) = barycentric quotient X(i)/X(j) for these {i,j}: {25, 1494}, {30, 305}, {669, 14380}, {1495, 69}, {1636, 4143}, {1637, 3267}, {1784, 561}, {1973, 2349}, {1974, 74}, {1990, 76}, {2173, 304}, {2420, 4563}, {2489, 2394}, {2971, 12079}, {3284, 3926}, {4240, 670}, {9406, 63}, {9407, 3}, {9408, 11064}, {9409, 3265}, {14398, 525}

X(14582) =  BARYCENTRIC PRODUCT X(125)*X(476)

Barycentrics    (b - c)*(b + c)*(a^2 - a*b + b^2 - c^2)*(a^2 + a*b + b^2 - c^2)*(-a^2 + b^2 - a*c - c^2)*(-a^2 + b^2 + a*c - c^2)*(-a^2 + b^2 + c^2) : :

X(14582) lies on these lines: {6, 2623}, {51, 512}, {53, 2501}, {94, 9979}, {216, 647}, {265, 10097}, {288, 2413}, {343, 525}, {476, 2395}, {1637, 1989}

X(14582) = isogonal conjugate of X(14590)
X(14582) = X(i)-cross conjugate of X(j) for these (i,j): {686, 647}, {3269, 11079}, {9409, 523}
X(14582) = cevapoint of X(1637) and X(12077)
X(14582) = crosspoint of X(94) and X(476)
X(14582) = crossdifference of every pair of points on line {186, 323}
X(14582) = crosssum of X(i) and X(j) for these (i,j): {50, 526}, {323, 8552}
X(14582) = X(i)-isoconjugate of X(j) for these (i,j): {19, 10411}, {50, 811}, {162, 323}, {163, 340}, {186, 662}, {648, 6149}, {4575, 14165}
X(14582) = barycentric product of Jerabek hyperbola intercepts of Hatzipolakis axis
X(14582) = barycentric product X(i)*X(j) for these {i,j}: {3, 10412}, {94, 647}, {125, 476}, {265, 523}, {328, 512}, {339, 14560}, {520, 6344}, {525, 1989}, {656, 2166}, {879, 14356}, {1141, 6368}, {3267, 11060}, {5627, 9033}, {14254, 14380}
X(14582) = barycentric quotient X(i)/X(j) for these {i,j}: {3, 10411}, {94, 6331}, {125, 3268}, {265, 99}, {328, 670}, {512, 186}, {523, 340}, {525, 7799}, {647, 323}, {810, 6149}, {878, 14355}, {1989, 648}, {2166, 811}, {2501, 14165}, {3049, 50}, {3269, 8552}, {6140, 2914}, {6344, 6528}, {6368, 1273}, {9033, 6148}, {9409, 1511}, {10412, 264}, {11060, 112}, {14270, 3043}, {14356, 877}, {14560, 250}

X(14583) =  X(2)X(476)∩X(51)X(512)

Barycentrics    (a^2 - a*b + b^2 - c^2)*(a^2 + a*b + b^2 - c^2)*(a^2 - b^2 - a*c + c^2)*(a^2 - b^2 + a*c + c^2)*(2*a^4 - a^2*b^2 - b^4 - a^2*c^2 + 2*b^2*c^2 - c^4) : :

X(14583) lies on these lines: {2, 476}, {4, 5627}, {13, 3441}, {14, 3440}, {25, 1989}, {30, 14254}, {51, 512}, {94, 14458}, {184, 14560}, {265, 541}, {1138, 14385}, {1495, 3081}, {2052, 6344}, {11074, 11080}

X(14583) = X(i)-Ceva conjugate of X(j) for these (i,j): {476, 1637}, {5627, 1989}, {6344, 1990}
X(14583) = X(i)-cross conjugate of X(j) for these (i,j): {9408, 1990}, {9409, 14560}
X(14583) = crosspoint of X(i) and X(j) for these (i,j): {30, 11070}, {1989, 5627}
X(14583) = crossdifference of every pair of points on line {323, 8552}
X(14583) = crosssum of X(323) and X(1511)
X(14583) = Danneels point of X(476)
X(14583) = X(i)-isoconjugate of X(j) for these (i,j): {75, 14385}, {323, 2349}, {1494, 6149}, {2159, 7799}
X(14583) = barycentric product X(i)*X(j) for these {i,j}: {6, 14254}, {30, 1989}, {94, 1495}, {265, 1990}, {476, 1637}, {2166, 2173}, {2420, 10412}, {3163, 5627}, {3260, 11060}, {3284, 6344}, {10272, 11071}
X(14583) = barycentric quotient X(i)/X(j) for these {i,j}: {30, 7799}, {32, 14385}, {1495, 323}, {1637, 3268}, {1989, 1494}, {1990, 340}, {2420, 10411}, {3163, 6148}, {9406, 6149}, {9407, 50}, {9408, 1511}, {9409, 8552}, {11060, 74}, {14254, 76}, {14398, 526}

X(14584) =  X(1)X(5)∩X(65)X(513)

Barycentrics    (2*a - b - c)*(a + b - c)*(a - b + c)*(a^2 - a*b + b^2 - c^2)*(a^2 - b^2 - a*c + c^2) : :

X(14584) lies on the cubics K054 and K360, and on these lines: {1, 5}, {7, 679}, {8, 765}, {44, 4530}, {56, 2222}, {65, 513}, {143, 5903}, {1168, 2099}, {1319, 1647}, {2082, 2161}, {10703, 12764}

X(14584) = X(1635)-zayin conjugate of X(654)
X(14584) = X(3259)-cross conjugate of X(900) X(14584) = crossdifference of every pair of points on line {654, 2323}
X(14584) = {X(14562),X(14564)}-harmonic conjugate of X(65)
X(14584) = X(i)-isoconjugate of X(j) for these (i,j): {36, 1320}, {88, 2323}, {106, 4511}, {214, 1318}, {654, 3257}, {901, 3738}, {903, 2361}, {1168, 4996}, {2316, 3218}, {3960, 5548}, {4080, 4282}, {4555, 8648}, {4997, 7113}
X(14584) = barycentric product X(i)*X(j) for these {i,j}: {80, 3911}, {519, 2006}, {655, 900}, {1411, 4358}, {2222, 3762}
X(14584) = barycentric quotient X(i)/X(j) for these {i,j}: {44, 4511}, {80, 4997}, {655, 4555}, {900, 3904}, {902, 2323}, {1319, 3218}, {1404, 36}, {1411, 88}, {1635, 3738}, {1960, 654}, {2006, 903}, {2161, 1320}, {2222, 3257}, {2251, 2361}, {3911, 320}, {6187, 2316}, {8756, 5081}

X(14585) = X(3)X(248)∩X(6)X(24)

Barycentrics    a^6*(a^2 - b^2 - c^2)^2 : :

X(14585) lies on these lines: {3, 248}, {4, 1970}, {6, 24}, {32, 184}, {39, 8779}, {49, 10317}, {112, 1614}, {125, 7749}, {154, 2207}, {156, 1625}, {185, 187}, {206, 2211}, {230, 6146}, {232, 10282}, {249, 7782}, {265, 8571}, {287, 1078}, {577, 1092}, {578, 10311}, {1147, 3289}, {1181, 3053}, {1204, 5206}, {1495, 3199}, {1562, 7756}, {1691, 6776}, {1692, 6467}, {1968, 3331}, {2715, 11257}, {3172, 9408}, {3202, 9419}, {3520, 13509}, {5023, 10605}, {7755, 10619}, {8743, 9707}, {10110, 10985}

X(14585) = isogonal conjugate of X(18027)
X(14585) = X(14533)-Ceva conjugate of X(184)
X(14585) = crosspoint of X(i) and X(j) for these (i,j): {6, 2351}, {184, 577}
X(14585) = crossdifference of every pair of points on line {850, 6368}
X(14585) = crosssum of X(i) and X(j) for these (i,j): {2, 317}, {264, 2052}
X(14585) = barycentric square of X(48)
X(14585) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (32, 184, 217), (54, 10312, 6), (1147, 10316, 3289), (1968, 6759, 3331), (1970, 1971, 4), (8779, 13367, 39)
X(14585) = X(i)-isoconjugate of X(j) for these (i,j): {4, 1969}, {69, 6521}, {75, 2052}, {76, 158}, {92, 264}, {273, 7017}, {304, 1093}, {305, 6520}, {318, 331}, {349, 1896}, {393, 561}, {823, 850}, {1096, 1502}, {1577, 6528}, {1928, 2207}, {8795, 14213}
X(14585) = barycentric product X(i)*X(j) for these {i,j}: {3, 184}, {6, 577}, {19, 4100}, {25, 1092}, {31, 255}, {32, 394}, {41, 7125}, {48, 48}, {54, 418}, {55, 7335}, {56, 6056}, {58, 4055}, {63, 9247}, {97, 217}, {154, 14379}, {163, 822}, {212, 603}, {216, 14533}, {228, 1437}, {248, 3289}, {326, 560}, {520, 1576}, {563, 1820}, {604, 2289}, {605, 606}, {810, 4575}, {1147, 2351}, {1259, 1397}, {1333, 3990}, {1409, 2193}, {1501, 3926}, {1790, 2200}, {1802, 7099}, {1804, 2175}, {1973, 6507}, {1974, 3964}, {2188, 7114}, {2206, 3682}, {3049, 4558}, {3917, 10547}, {6394, 9418}, {6413, 8911}, {7055, 9448}, {7183, 9447}
X(14585) = barycentric quotient X(i)/X(j) for these {i,j}: {32, 2052}, {48, 1969}, {184, 264}, {217, 324}, {255, 561}, {326, 1928}, {394, 1502}, {418, 311}, {560, 158}, {577, 76}, {1092, 305}, {1501, 393}, {1576, 6528}, {1917, 1096}, {1973, 6521}, {1974, 1093}, {4055, 313}, {4100, 304}, {6056, 3596}, {7335, 6063}, {9233, 2207}, {9247, 92}, {9418, 6530}, {9448, 1857}, {14533, 276}

X(14586) =  BARYCENTRIC PRODUCT X(54)*X(110)

Barycentrics    a^4*(a - b)*(a + b)*(a - c)*(a + c)*(a^4 - 2*a^2*b^2 + b^4 - a^2*c^2 - b^2*c^2)*(a^4 - a^2*b^2 - 2*a^2*c^2 - b^2*c^2 + c^4) : :

X(14586) lies on these lines: {6, 11077}, {39, 54}, {112, 933}, {115, 1141}, {187, 1157}, {252, 7749}, {1625, 2623}, {2148, 2150}, {3484, 13509}, {8565, 10274}

X(14586) = isogonal conjugate of X(18314)
X(14586) = isotomic conjugate of X(15415)
X(14586) = X(i)-cross conjugate of X(j) for these (i,j): {1576, 933}, {14270, 1141}
X(14586) = cevapoint of X(i) and X(j) for these (i,j): {6, 2623}, {570, 647}, {1879, 12077}
X(14586) = trilinear pole of line X(160)X(184)
X(14586) = crosssum of X(6368) and X(12077)
X(14586) = X(i)-isoconjugate of X(j) for these (i,j): {2, 2618}, {5, 1577}, {53, 14208}, {75, 12077}, {92, 6368}, {311, 661}, {324, 656}, {338, 2617}, {523, 14213}, {850, 1953}, {2181, 3267}
X(14586) = barycentric product of circumcircle intercepts of line X(3)X(54)
X(14586) = barycentric product X(i)*X(j) for these {i,j}: {3, 933}, {54, 110}, {95, 1576}, {97, 112}, {162, 2169}, {163, 2167}, {249, 2623}, {648, 14533}, {662, 2148}, {1101, 2616}, {1157, 1291}, {2190, 4575}, {4558, 8882}, {8883, 13398}
X(14586) = barycentric quotient X(i)/X(j) for these {i,j}: {31, 2618}, {32, 12077}, {54, 850}, {97, 3267}, {110, 311}, {112, 324}, {163, 14213}, {184, 6368}, {933, 264}, {1576, 5}, {2148, 1577}, {2169, 14208}, {2623, 338}, {14533, 525}

X(14587) = X(49)-CROSS CONJUGATE OF X(110)

Barycentrics    a^4*(a - b)^2*(a + b)^2*(a - c)^2*(a + c)^2*(a^4 - 2*a^2*b^2 + b^4 - a^2*c^2 - b^2*c^2)*(a^4 - a^2*b^2 - 2*a^2*c^2 - b^2*c^2 + c^4) : :

X)14587) lies on these lines: {250, 2070}, {933, 10420}

X(14587) = X(49)-cross conjugate of X(110)
X(14587) = X(i)-isoconjugate of X(j) for these (i,j): {5, 1109}, {115, 14213}, {137, 2962}, {311, 2643}, {324, 3708}, {338, 1953}, {339, 2181}, {523, 2618}, {1087, 8901}, {1577, 12077}, {2632, 13450}
X(14587) = cevapoint of X(i) and X(j) for these (i,j): {54, 110}, {1576, 2965}
X(14587) = barycentric product X(i)*X(j) for these {i,j}: {54, 249}, {97, 250}, {933, 4558}, {1101, 2167}
X(14587) = barycentric quotient X(i)/X(j) for these {i,j}: {54, 338}, {97, 339}, {163, 2618}, {249, 311}, {250, 324}, {1101, 14213}, {1576, 12077}, {2148, 1109}, {2965, 137}, {8882, 2970}, {14533, 125}

X(14588) = X(99)-WAW CONJUGATE OF X(2)

Barycentrics    (a - b)*(a + b)*(a - c)*(a + c)*(2*a^4 - 2*a^2*b^2 + b^4 - 2*a^2*c^2 + c^4) : :
5 X(99) + X(892), X(892) - 5 X(4590), X(671) - 4 X(9164), 2 X(892) - 5 X(9182), 2 X(99) + X(9182)

X(14588) lies on these lines: {23, 325}, {99, 523}, {110, 2858}, {524, 2076}, {671, 9164}, {1510, 12833}, {2396, 5467}

X(14588) = midpoint of X(99) and X(4590)
X(14588) = reflection of X(9182) in X(4590)
X(14588) = X(11123)-cross conjugate of X(620)
X(14588) = cevapoint of X(620) and X(11123)
X(14588) = X(99)-daleth conjugate of X(4590)
X(14588) = X(99)-waw conjugate of X(2)
X(14588) = X(i)-Ceva conjugate of X(j) for these (i,j): {99, 11123}, {671, 5468}, {9164, 9182}
X(14588) = barycentric product X(i)*X(j) for these {i,j}: {99, 620}, {4590, 11123}
X(14588) = barycentric quotient X(i)/X(j) for these {i,j}: {620, 523}, {11123, 115}

X(14589) = X(100)X(650)∩X(101)X(2743)

Barycentrics    a*(a - b)*(a - c)*(2*a^3 - 2*a^2*b - a*b^2 + b^3 - 2*a^2*c + 4*a*b*c - b^2*c - a*c^2 - b*c^2 + c^3) : :

X(14589) lies on these lines: {44, 3684}, {55, 5701}, {100, 650}, {101, 2743}, {901, 4790}, {910, 5537}, {4564, 9358}, {4885, 4998}, {6603, 9356}

X(14589) = X(i)-Ceva conjugate of X(j) for these (i,j): {100, 11124}, {294, 2284}
X(14589) = X(11124)-cross conjugate of X(3035)
X(14589) = X(100)-waw conjugate of X(1376)
X(14589) = X(2170)-zayin conjugate of X(513)
X(14589) = {X(100),X(1252)}-harmonic conjugate of X(650)
X(14589) = barycentric product X(i)*X(j) for these {i,j}: {100, 3035}, {4998, 11124}
X(14589) = barycentric quotient X(i)/X(j) for these {i,j}: {3035, 693}, {11124, 11}

X(14590) =  BARYCENTRIC QUOTIENT X(50)/X(647)

Barycentrics    a^2*(a - b)*(a + b)*(a - c)*(a + c)*(a^2 + b^2 - c^2)*(a^2 - b^2 - b*c - c^2)*(a^2 - b^2 + b*c - c^2)*(a^2 - b^2 + c^2) : :

X(14590) lies on these lines: {2, 95}, {4, 10414}, {50, 340}, {99, 112}, {250, 4230}, {691, 10098}, {1304, 10420}, {3098, 8541}

X(14590) = isogonal conjugate of X(14582)
X(14590) = isotomic conjugate of X(14592)
X(14590) = X(687)-Ceva conjugate of X(648)
X(14590) = X(i)-cross conjugate of X(j) for these (i,j): {526, 340}, {8552, 323}
X(14590) = cevapoint of X(i) and X(j) for these (i,j): {50, 526}, {323, 8552}
X(14590) = trilinear pole of line X(186)X(323)
X(14590) = crosssum of X(1637) and X(12077)
X(14590) = {X(4230),X(5467)}-harmonic conjugate of X(250)
X(14590) = X(i)-isoconjugate of X(j) for these (i,j): {48, 10412}, {94, 810}, {265, 661}, {328, 798}, {476, 3708}, {647, 2166}, {656, 1989}, {822, 6344}, {2618, 11077}, {2631, 5627}, {11060, 14208}
X(14590) = barycentric product X(i)*X(j) for these {i,j}: {4, 10411}, {50, 6331}, {99, 186}, {110, 340}, {112, 7799}, {250, 3268}, {323, 648}, {811, 6149}, {877, 14355}, {933, 1273}, {1304, 6148}, {4558, 14165}
X(14590) = barycentric quotient X(i)/X(j) for these {i,j}: {4, 10412}, {50, 647}, {99, 328}, {107, 6344}, {110, 265}, {112, 1989}, {162, 2166}, {186, 523}, {250, 476}, {323, 525}, {340, 850}, {526, 125}, {648, 94}, {933, 1141}, {1154, 6368}, {1304, 5627}, {1511, 9033}, {2624, 3708}, {3043, 526}, {3268, 339}, {4230, 14356}, {4240, 14254}, {4242, 6757}, {6149, 656}, {7799, 3267}, {10411, 69}, {10420, 12028}, {11062, 12077}, {14355, 879}, {14385, 14380}

X(14591) = X(6)X(24)∩X(110)X(112)

Barycentrics    a^4*(a - b)*(a + b)*(a - c)*(a + c)*(a^2 + b^2 - c^2)*(a^2 - b^2 - b*c - c^2)*(a^2 - b^2 + b*c - c^2)*(a^2 - b^2 + c^2) : :

X(14591) lies on these lines: {6, 24}, {50, 14385}, {110, 112}, {186, 2088}, {249, 4235}, {879, 935}

X(14591) = isogonal conjugate of X(14592)
X(14591) = X(i)-Ceva conjugate of X(j) for these (i,j): {249, 1986}, {1304, 1576}
X(14591) = X(14270)-cross conjugate of X(186)
X(14591) = crossdifference of every pair of points on line {125, 6368}
X(14591) = X(i)-isoconjugate of X(j) for these (i,j): {63, 10412}, {94, 656}, {265, 1577}, {328, 661}, {525, 2166}, {1989, 14208}
X(14591) = barycentric product X(i)*X(j) for these {i,j}: {25, 10411}, {50, 648}, {110, 186}, {112, 323}, {162, 6149}, {250, 526}, {340, 1576}, {476, 3043}, {933, 1154}, {1291, 2914}, {1304, 1511}, {1986, 10420}, {4230, 14355}, {4240, 14385}, {11702, 13863}
X(14591) = barycentric quotient X(i)/X(j) for these {i,j}: {25, 10412}, {50, 525}, {110, 328}, {112, 94}, {186, 850}, {323, 3267}, {526, 339}, {1576, 265}, {3043, 3268}, {6149, 14208}, {10411, 305}, {14270, 125}

X(14592) =  BARYCENTRIC PRODUCT X(94)*X(525)

Barycentrics    b^2*(b - c)*c^2*(b + c)*(a^2 - a*b + b^2 - c^2)*(a^2 + a*b + b^2 - c^2)*(-a^2 + b^2 - a*c - c^2)*(-a^2 + b^2 + a*c - c^2)*(-a^2 + b^2 + c^2) : :

X(14592) lies on these lines: {2, 2413}, {5, 523}, {94, 2394}, {265, 879}, {343, 525}, {476, 935}, {850, 5664}

X(14592) = isogonal conjugate of X(14591)
X(14592) = isotomic conjugate of X(14590)
X(14592) = X(i)-cross conjugate of X(j) for these (i,j): {115, 12028}, {6334, 525}, {9033, 850}
X(14592) = X(i)-isoconjugate of X(j) for these (i,j): {50, 162}, {112, 6149}, {163, 186}, {250, 2624}, {933, 2290}, {1973, 10411}
X(14592) = trilinear pole of line X(125)X(6368)
X(14592) = barycentric product X(i)*X(j) for these {i,j}: {69, 10412}, {94, 525}, {265, 850}, {328, 523}, {339, 476}, {1989, 3267}, {2166, 14208}, {3265, 6344}
X(14592) = barycentric quotient X(i)/X(j) for these {i,j}: {69, 10411}, {94, 648}, {125, 526}, {265, 110}, {328, 99}, {339, 3268}, {476, 250}, {523, 186}, {525, 323}, {526, 3043}, {647, 50}, {656, 6149}, {850, 340}, {879, 14355}, {1141, 933}, {1989, 112}, {2166, 162}, {3267, 7799}, {3708, 2624}, {5627, 1304}, {6344, 107}, {6368, 1154}, {6757, 4242}, {9033, 1511}, {10412, 4}, {12028, 10420}, {12077, 11062}, {14254, 4240}, {14356, 4230}, {14380, 14385}

X(14593) = X(4)X(52)∩X(25)X(53)

Barycentrics    (a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*(a^4 - 2*a^2*b^2 + b^4 - 2*b^2*c^2 + c^4)*(a^4 + b^4 - 2*a^2*c^2 - 2*b^2*c^2 + c^4) : :

The trilinear polar of X(14593) passes through X(2489). (Randy Hutson, November 2, 2017)

X(14593) lies on the cubic K350 and these lines: {2, 136}, {4, 52}, {5, 8906}, {25, 53}, {96, 7487}, {184, 8754}, {254, 1147}, {421, 11547}, {428, 14486}, {460, 2207}, {485, 8940}, {486, 8944}, {1899, 2970}, {5200, 13429}, {7745, 9777}, {8800, 9937}

X(14593) = isogonal conjugate of X(9723)
X(14593) = X(i)-Ceva conjugate of X(j) for these (i,j): {847, 2165}, {925, 2501}
X(14593) = X(i)-cross conjugate of X(j) for these (i,j): {32, 393}, {51, 25}
X(14593) = cevapoint of X(i) and X(j) for these (i,j): {512, 8754}, {8576, 8577}
X(14593) = crosssum of X(3) and X(6503)
X(14593) = {X(4),X(847)}-harmonic conjugate of X(68)
X(14593) = X(i)-isoconjugate of X(j) for these (i,j): {1, 9723}, {24, 326}, {47, 69}, {48, 7763}, {63, 1993}, {75, 1147}, {76, 563}, {255, 317}, {304, 571}, {394, 1748}, {924, 4592}, {1102, 8745}, {4575, 6563}, {6507, 11547}
X(14593) = barycentric product X(i)*X(j) for these {i,j}: {4, 2165}, {6, 847}, {19, 91}, {25, 5392}, {53, 96}, {68, 393}, {158, 1820}, {925, 2501}, {1989, 5962}, {2052, 2351}
X(14593) = barycentric quotient X(i)/X(j) for these {i,j}: {4, 7763}, {6, 9723}, {25, 1993}, {32, 1147}, {68, 3926}, {91, 304}, {393, 317}, {560, 563}, {847, 76}, {925, 4563}, {1096, 1748}, {1820, 326}, {1973, 47}, {1974, 571}, {2165, 69}, {2207, 24}, {2351, 394}, {2489, 924}, {2501, 6563}, {3199, 52}, {5392, 305}, {5962, 7799}, {6524, 11547}, {8576, 5408}, {8577, 5409}, {11060, 5961}

X(14594) =  X(100)X(108)∩X(109)X(190)

Barycentrics    (a - b)*(a - c)*(a + b - c)*(a - b + c)*(a^2 + b^2 + 2*b*c + c^2) : :

X(14594) lies on these lines: {100, 108}, {109, 190}, {210, 1943}, {312, 8270}, {651, 3952}, {664, 668}, {726, 9364}, {833, 2222}, {927, 4624}, {1038, 4385}, {1214, 7081}, {1441, 5297}, {1456, 4009}, {1465, 5205}, {1708, 3769}, {1758, 4434}, {3701, 4296}, {3891, 5435}, {4318, 4358}

X(14594) = X(2517)-cross conjugate of X(1010)
X(14594) = cevapoint of X(612) and X(6590) X(14594) = {X(664),X(3699)}-harmonic conjugate of X(4551)
X(14594) = trilinear pole of line X(388)X(2285)
X(14594) = X(i)-isoconjugate of X(j) for these (i,j): {513, 1036}, {522, 1472}, {649, 2339}, {650, 2221}, {1039, 1459}, {1245, 3737}, {1310, 3271}, {2281, 4560}
X(14594) = barycentric product X(i)*X(j) for these {i,j}: {190, 388}, {612, 4554}, {646, 4320}, {651, 4385}, {658, 3974}, {664, 2345}, {668, 2285}, {1010, 4552}, {1038, 6335}, {1460, 1978}, {2517, 4564}, {3699, 7365}, {4033, 5323}, {4998, 6590}, {6558, 7197}, {7257, 8898}, {7258, 10376}
X(14594) = barycentric quotient X(i)/X(j) for these {i,j}: {100, 2339}, {101, 1036}, {109, 2221}, {388, 514}, {612, 650}, {1010, 4560}, {1038, 905}, {1415, 1472}, {1460, 649}, {1783, 1039}, {2285, 513}, {2286, 1459}, {2303, 3737}, {2345, 522}, {2484, 3271}, {2517, 4858}, {2522, 7004}, {3974, 3239}, {4320, 3669}, {4385, 4391}, {4559, 1245}, {4564, 1310}, {5227, 521}, {5323, 1019}, {6590, 11}, {7085, 652}, {7102, 3064}, {7365, 3676}, {8678, 2170}, {8898, 4017}, {10376, 7216}

X(14595) =  X(265)X(1531)∩X(1495)X(1989)

Barycentrics    (a^2 - b^2 - c^2)*(a^2 - a*b + b^2 - c^2)^2*(a^2 + a*b + b^2 - c^2)^2*(a^2 - b^2 - a*c + c^2)^2*(a^2 - b^2 + a*c + c^2)^2 : :

X(14595) lies on these lines: {265, 1531}, {1495, 1989}, {5627, 7687}, {11074, 11080}, {11581, 14373}, {11582, 14372}

X(14595) = X(i)-isoconjugate of X(j) for these (i,j): {75, 3043}, {340, 6149}
X(14595) = barycentric product X(i)*X(j) for these {i,j}: {265, 1989}, {328, 11060}, {10217, 11085}, {10218, 11080}, {11079, 14254}
X(14595) = barycentric quotient X(i)/X(j) for these {i,j}: {32, 3043}, {265, 7799}, {1989, 340}, {10217, 11128}, {10218, 11129}, {11060, 186}

X(14596) =  BARYCENTRIC PRODUCT X(7)*X(177)

Barycentrics    Sqrt(a)*(a + b - c)*(a - b + c)*(Sqrt(b*(a + b - c))*(a - b + c) + (a + b - c)*Sqrt(c*(a - b + c))) : :

X(14596) lies on the cubic K746, the circumconic {A,B,C,X(1)X(2)}, and these lines: {1, 167}, {2, 4146}, {57, 7371}, {105, 13444}

X(14596) = X(7371)-Ceva conjugate of X(10490)
X(14596) = X(10490)-cross conjugate of X(2091)
X(14596) = crosspoint of X(279) and X(7371)
X(14596) = crosssum of X(220) and X(6726)
X(14596) = X(i)-isoconjugate of X(j) for these (i,j): {9, 260}, {3939, 10492}
X(14596) = barycentric product X(i)*X(j) for these {i,j}: {7, 177}, {174, 234}, {178, 7371}, {555, 7707}, {693, 13444}, {2091, 7057}, {4146, 10490}
X(14596) = barycentric quotient X(i)/X(j) for these {i,j}: {56, 260}, {177, 8}, {178, 7027}, {234, 556}, {2091, 7048}, {3669, 10492}, {7707, 6731}, {10490, 188}, {13444, 100}

X(14597) = CROSSSUM OF X(4) AND X(37)

Barycentrics    a^3*(a^2 - b^2 - c^2)*(a^2*b - b^3 + a^2*c + 2*a*b*c + b^2*c + b*c^2 - c^3) : :

X(14597) lies on these lines: {3, 3990}, {6, 57}, {37, 1071}, {48, 577}, {71, 3917}, {73, 216}, {213, 2178}, {219, 3916}, {604, 2288}, {849, 1333}, {942, 1841}, {1364, 2269}, {1950, 2302}, {2197, 2252}

X(14597) = crosspoint of X(i) and X(j) for these (i,j): {3, 81}, {222, 1437}
X(14597) = crossdifference of every pair of points on line {3900, 4036}
X(14597) = crosssum of X(i) and X(j) for these (i,j): {4, 37}, {9, 3191}
X(14597) = {X(48),X(603)}-harmonic conjugate of X(577)
X(14597) = X(i)-isoconjugate of X(j) for these (i,j): {92, 943}, {264, 2259}, {318, 2982}, {1794, 2052}
X(14597) = barycentric product X(i)*X(j) for these {i,j}: {1, 4303}, {3, 942}, {48, 5249}, {63, 2260}, {77, 14547}, {255, 1838}, {394, 1841}, {442, 1437}, {500, 7100}, {603, 6734}, {1439, 8021}, {1790, 2294}, {1804, 1859}
X(14597) = barycentric quotient X(i)/X(j) for these {i,j}: {184, 943}, {942, 264}, {1841, 2052}, {2260, 92}, {4303, 75}, {5249, 1969}, {9247, 2259}, {14547, 318}

X(14598) =  BARYCENTRIC PRODUCT X(1)*X(1922)

Barycentrics    a^5*(-b^2 + a*c)*(a*b - c^2) : :

X(14598) lies on these lines: {31, 292}, {32, 1922}, {58, 291}, {171, 334}, {692, 1333}, {713, 813}, {741, 785}, {7121, 10799}, {8022, 9468}

X(14598) = crosssum of X(1921) and X(4087)
X(14598) = X(2175)-beth conjugate of X(692)
X(14598) = X(9455)-cross conjugate of X(560)
X(14598) = X(i)-isoconjugate of X(j) for these (i,j): {2, 1921}, {7, 4087}, {75, 350}, {76, 239}, {85, 3975}, {238, 561}, {242, 305}, {256, 1926}, {257, 3978}, {274, 3948}, {310, 740}, {312, 10030}, {659, 6386}, {668, 3766}, {670, 4010}, {693, 874}, {812, 1978}, {871, 3783}, {1447, 3596}, {1502, 1914}, {1928, 2210}, {1966, 7018}, {2238, 6385}, {3261, 3570}, {3685, 6063}, {3716, 4572}, {4455, 4609}, {4594, 14295}
X(14598) = X(i)-Hirst inverse of X(j) for these (i,j): {32, 1922}, {1911, 7122}
X(14598) = trilinear pole of line X(1980)X(2205)
X(14598) = barycentric product X(i)*X(j) for these {i,j}: {1, 1922}, {6, 1911}, {25, 2196}, {31, 292}, {32, 291}, {101, 875}, {171, 9468}, {172, 1967}, {213, 741}, {295, 1973}, {334, 1501}, {335, 560}, {604, 7077}, {660, 1919}, {667, 813}, {669, 4584}, {692, 3572}, {694, 7122}, {894, 1927}, {1397, 4876}, {1402, 2311}, {1909, 8789}, {1924, 4589}, {1977, 5378}, {1980, 4562}, {4639, 9426}, {7233, 9447}
X(14598) = barycentric quotient X(i)/X(j) for these {i,j}: {31, 1921}, {32, 350}, {41, 4087}, {172, 1926}, {291, 1502}, {292, 561}, {335, 1928}, {560, 239}, {741, 6385}, {813, 6386}, {875, 3261}, {1397, 10030}, {1501, 238}, {1911, 76}, {1917, 1914}, {1918, 3948}, {1919, 3766}, {1922, 75}, {1924, 4010}, {1927, 257}, {1980, 812}, {2175, 3975}, {2196, 305}, {2205, 740}, {4584, 4609}, {7122, 3978}, {8630, 4486}, {8789, 256}, {9233, 2210}, {9447, 3685}, {9448, 3684}, {9468, 7018}

X(14599) =  CROSSSUM OF X(335) AND X(337)

Barycentrics    a^4*(a^2 - b*c) : :

X(14599) lies on these lines: {6, 7295}, {31, 1501}, {32, 560}, {41, 3774}, {42, 5371}, {171, 1915}, {238, 1691}, {584, 2273}, {849, 1333}, {1110, 2251}, {1197, 2194}, {1692, 3271}, {1914, 5009}, {1919, 8637}, {1922, 9455}, {2076, 3792}

X(14599) = isogonal conjugate of X(18895)
X(14599) = crossdifference of every pair of points on line {313, 3261}
X(14599) = crosssum of X(i) and X(j) for these (i,j): {321, 3263}, {335, 337}
X(14599) = X(32)-Hirst inverse of X(560)
X(14599) = X(i)-isoconjugate of X(j) for these (i,j): {2, 334}, {75, 335}, {76, 291}, {85, 4518}, {92, 337}, {292, 561}, {295, 1969}, {312, 7233}, {514, 4583}, {523, 4639}, {660, 3261}, {668, 4444}, {693, 4562}, {850, 4584}, {871, 3862}, {876, 1978}, {894, 1934}, {1502, 1911}, {1577, 4589}, {1581, 1920}, {1909, 1916}, {1922, 1928}, {3572, 6386}, {4876, 6063}
X(14599) = barycentric product X(i)*X(j) for these {i,j}: {1, 2210}, {6, 1914}, {25, 7193}, {31, 238}, {32, 239}, {41, 1429}, {42, 5009}, {48, 2201}, {55, 1428}, {58, 3747}, {101, 8632}, {110, 4455}, {184, 242}, {256, 1933}, {350, 560}, {385, 7104}, {604, 3684}, {659, 692}, {667, 3573}, {740, 2206}, {862, 1437}, {874, 1980}, {893, 1691}, {904, 1580}, {1284, 2194}, {1333, 2238}, {1397, 3685}, {1408, 4433}, {1415, 4435}, {1447, 2175}, {1501, 1921}, {1576, 4010}, {1911, 8300}, {1919, 3570}, {1922, 4366}, {6654, 9454}, {7077, 12835}, {9447, 10030}
X(14599) = barycentric quotient X(i)/X(j) for these {i,j}: {31, 334}, {32, 335}, {163, 4639}, {184, 337}, {238, 561}, {239, 1502}, {350, 1928}, {560, 291}, {692, 4583}, {904, 1934}, {1397, 7233}, {1428, 6063}, {1501, 292}, {1576, 4589}, {1691, 1920}, {1914, 76}, {1917, 1911}, {1919, 4444}, {1933, 1909}, {1980, 876}, {2175, 4518}, {2201, 1969}, {2210, 75}, {3573, 6386}, {3747, 313}, {4455, 850}, {5009, 310}, {7104, 1916}, {7193, 305}, {8632, 3261}, {9233, 1922}, {9447, 4876}, {9448, 7077}

X(14600) = BARYCENTRIC PRODUCT X(6)*X(248)

Barycentrics    a^4*(a^2 - b^2 - c^2)*(a^4 + b^4 - a^2*c^2 - b^2*c^2)*(a^4 - a^2*b^2 - b^2*c^2 + c^4) : :

X(14600) lies on these lines: {3, 248}, {6, 3425}, {25, 1501}, {32, 2909}, {98, 230}, {217, 10547}, {287, 343}, {1576, 9419}, {2021, 3455}, {3094, 14355}, {3504, 6638}, {5621, 11653}, {6531, 8884}

X(14600) = X(2715)-Ceva conjugate of X(878)
X(14600) = crosspoint of X(248) and X(1976)
X(14600) = crosssum of X(297) and X(325)
X(14600) = {X(1691),X(1971)}-harmonic conjugate of X(1513)
X(14600) = X(i)-isoconjugate of X(j) for these (i,j): {75, 297}, {76, 240}, {92, 325}, {158, 6393}, {232, 561}, {264, 1959}, {304, 6530}, {511, 1969}, {811, 2799}, {823, 6333}, {877, 1577}, {1235, 3405}, {1928, 2211}
X(14600) = barycentric product X(i)*X(j) for these {i,j}: {3, 1976}, {6, 248}, {31, 293}, {32, 287}, {48, 1910}, {98, 184}, {110, 878}, {336, 560}, {577, 6531}, {647, 2715}, {879, 1576}, {1821, 9247}, {1974, 6394}, {2422, 4558}, {2966, 3049}
X(14600) = barycentric quotient X(i)/X(j) for these {i,j}: {32, 297}, {184, 325}, {248, 76}, {287, 1502}, {293, 561}, {336, 1928}, {560, 240}, {577, 6393}, {878, 850}, {1501, 232}, {1576, 877}, {1910, 1969}, {1974, 6530}, {1976, 264}, {2715, 6331}, {3049, 2799}, {9233, 2211}, {9247, 1959}, {9418, 2967}

X(14601) = BARYCENTRIC PRODUCT X(32)*X(98)

Barycentrics    a^4*(a^4 + b^4 - a^2*c^2 - b^2*c^2)*(a^4 - a^2*b^2 - b^2*c^2 + c^4) : :

X(14601) lies on these lines: {5, 83}, {6, 157}, {32, 2909}, {184, 11672}, {290, 3114}, {399, 11653}, {460, 6531}, {729, 2715}, {1084, 1974}, {2207, 2971}, {2966, 3225}, {5967, 9516}, {9418, 9468}, {11610, 11641}

X(14601) = crosssum of X(325) and X(6393)
X(14601) = trilinear pole of line X(669)X(1501)
X(14601) = barycentric product of circumcircle intercepts of circle {{X(1687),X(1688),PU(1),PU(2)}}
X(14601) = X(i)-isoconjugate of X(j) for these (i,j): {75, 325}, {76, 1959}, {92, 6393}, {237, 1928}, {240, 305}, {297, 304}, {511, 561}, {799, 2799}, {811, 6333}, {877, 14208}, {1502, 1755}, {1577, 2396}, {1934, 5976}, {3405, 8024}, {3569, 4602}
X(14601) = barycentric product X(i)*X(j) for these {i,j}: {6, 1976}, {25, 248}, {31, 1910}, {32, 98}, {110, 2422}, {112, 878}, {184, 6531}, {287, 1974}, {290, 1501}, {293, 1973}, {512, 2715}, {560, 1821}, {669, 2966}, {685, 3049}, {1576, 2395}, {1662, 1663}, {1692, 2065}, {2353, 11610}, {8789, 14382}, {11060, 14355}
X(14601) = barycentric quotient X(i)/X(j) for these {i,j}: {32, 325}, {98, 1502}, {184, 6393}, {248, 305}, {560, 1959}, {669, 2799}, {878, 3267}, {1084, 868}, {1501, 511}, {1576, 2396}, {1821, 1928}, {1910, 561}, {1917, 1755}, {1974, 297}, {1976, 76}, {2422, 850}, {2715, 670}, {2966, 4609}, {3049, 6333}, {9233, 237}, {9426, 3569}

X(14602) = CROSSSUM OF X(2) AND X(5207)

Barycentrics    a^4*(a^2 - b*c)*(a^2 + b*c) : :

X(14602) lies on these lines: {6, 6660}, {23, 251}, {32, 184}, {39, 5012}, {76, 4159}, {110, 3229}, {187, 249}, {560, 7104}, {1691, 8623}, {1692, 1976}, {1915, 3398}, {3978, 4027}, {4121, 7826}, {9418, 9468}

X(14602) = crosspoint of X(251) and X(1976)
X(14602) = crossdifference of every pair of points on line {850, 2528}
X(14602) = crosssum of X(i) and X(j) for these (i,j): {2, 5207}, {141, 325}
X(14602) = X(32)-Hirst inverse of X(1501)
X(14602) = {X(32),X(184)}-harmonic conjugate of X(3117)
X(14602) = X(i)-isoconjugate of X(j) for these (i,j): {2, 1934}, {75, 1916}, {76, 1581}, {257, 334}, {335, 7018}, {561, 694}, {882, 4602}, {1502, 1967}, {1928, 9468}
X(14602) = barycentric product X(i)*X(j) for these {i,j}: {1, 1933}, {6, 1691}, {31, 1580}, {32, 385}, {110, 5027}, {171, 2210}, {172, 1914}, {184, 419}, {238, 7122}, {249, 2086}, {251, 8623}, {560, 1966}, {692, 4164}, {804, 1576}, {880, 9426}, {1428, 2330}, {1501, 3978}, {1917, 1926}, {1974, 12215}, {2206, 4039}, {4027, 9468}, {9418, 14382}
X(14602) = barycentric quotient X(i)/X(j) for these {i,j}: {31, 1934}, {32, 1916}, {385, 1502}, {560, 1581}, {1501, 694}, {1580, 561}, {1691, 76}, {1917, 1967}, {1933, 75}, {1966, 1928}, {2086, 338}, {2210, 7018}, {5027, 850}, {7122, 334}, {8623, 8024}, {9233, 9468}, {9426, 882}

X(14603) = ISOGONAL CONJUGATE OF X(8789)

Barycentrics    b^4*c^4*(-a^2 + b*c)*(a^2 + b*c) : :

X(14603) lies on these lines: {23, 689}, {76, 141}, {308, 3934}, {511, 670}, {561, 7018}, {732, 3978}, {880, 12215}, {3266, 4609}, {3932, 6386}, {5976, 8783}

X(14603) = isogonal conjugate of X(8789)
X(14603) = isotomic conjugate of X(9468)
X(14603) = X(i)-isoconjugate of X(j) for these (i,j): {1, 8789}, {6, 1927}, {31, 9468}, {32, 1967}, {163, 881}, {560, 694}, {733, 1923}, {805, 1924}, {904, 1922}, {1501, 1581}, {1911, 7104}, {1916, 1917}, {1934, 9233}
X(14603) = X(76)-Hirst inverse of X(1502)
X(14603) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (76, 6374, 3094), (1502, 10010, 76)
X(14603) = barycentric product X(i)*X(j) for these {i,j}: {75, 1926}, {76, 3978}, {385, 1502}, {561, 1966}, {670, 14295}, {804, 4609}, {850, 880}, {1580, 1928}, {1920, 1921}, {4087, 7205}, {6386, 14296}
X(14603) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 1927}, {2, 9468}, {6, 8789}, {75, 1967}, {76, 694}, {239, 7104}, {308, 733}, {325, 14251}, {350, 904}, {385, 32}, {419, 1974}, {523, 881}, {561, 1581}, {670, 805}, {732, 3051}, {804, 669}, {850, 882}, {880, 110}, {894, 1922}, {1502, 1916}, {1580, 560}, {1691, 1501}, {1909, 1911}, {1920, 292}, {1921, 893}, {1926, 1}, {1928, 1934}, {1933, 1917}, {1966, 31}, {2086, 9427}, {2236, 1923}, {3978, 6}, {4039, 1918}, {4107, 1919}, {4164, 1980}, {4374, 875}, {5027, 9426}, {5976, 237}, {8783, 6660}, {9865, 3117}, {12215, 184}, {14295, 512}, {14382, 1976}

X(14604) =  BARYCENTRIC PRODUCT X(6)X(8789)

Barycentrics    a^8*(-b^2 + a*c)*(b^2 + a*c)*(a*b - c^2)*(a*b + c^2) : :

X(14604) lies on these lines: {32, 694}, {251, 1916}, {1501, 8023}, {1576, 9468}, {7104, 9232}

X(14604) = X(1501)-Hirst inverse of X(8789)
X(14604) = X(i)-isoconjugate of X(j) for these (i,j): {76, 1926}, {385, 1928}, {561, 3978}, {1502, 1966}, {4602, 14295}
X(14604) = barycentric product X(i)*X(j) for these {i,j}: {6, 8789}, {31, 1927}, {32, 9468}, {560, 1967}, {694, 1501}, {805, 9426}, {881, 1576}, {1581, 1917}, {1916, 9233}, {1922, 7104}
X(14604) = barycentric quotient X(i)/X(j) for these {i,j}: {560, 1926}, {1501, 3978}, {1917, 1966}, {1927, 561}, {1967, 1928}, {8789, 76}, {9233, 385}, {9426, 14295}, {9468, 1502}

X(14605) =  X(3)X(67)∩X(74)X(9069)

Barycentrics    a^10-4 a^8 b^2-3 a^6 b^4+9 a^4 b^6-4 a^2 b^8+b^10-4 a^8 c^2+15 a^6 b^2 c^2-8 a^4 b^4 c^2-b^8 c^2-3 a^6 c^4-8 a^4 b^2 c^4+4 a^2 b^4 c^4+9 a^4 c^6-4 a^2 c^8-b^2 c^8+c^10 : :
X(14605) = 3 X(9759) - X(10706)

X(14605) lies on these lines: {3,67}, {74,9069}, {125,11287}, {1513,9759}, {5642,11288}, {5976,11006}, {8356,9140}

X(14606) = X(2)X(351)∩X(6)X(888)

Barycentrics    a^2*(b - c)*(b + c)*(a^4*b^2 + a^2*b^4 - 2*a^4*c^2 - 2*b^4*c^2 + a^2*c^4 + b^2*c^4)*(2*a^4*b^2 - a^2*b^4 - a^4*c^2 - b^4*c^2 - a^2*c^4 + 2*b^2*c^4) : :

X(14606) lies on the cubic K017, the circumconic {A, B, C, X(2), X(6)}, and these lines: {2, 351}, {6, 888}, {111, 669}, {263, 9135}, {512, 694}, {523, 3228}, {1084, 9178}, {3572, 4128}, {5652, 5969}

X(14606) = reflection of X(9178) in X(1084)
X(14606) = isogonal conjugate of X(14607)
X(14606) = crossdifference of every pair of points on line {5106, 5969}
X(14606) = crosssum of X(5969) and X(11182)
X(14606) = X(5652)-line conjugate of X(5969)
X(14606) = trilinear pole of line X(512)X(2086)
X(14606) = X(i)-isoconjugate of X(j) for these (i,j): {662, 5969}, {799, 5106}
X(14606) = barycentric product X(523)*X(5970)
X(14606) = barycentric quotient X(i)/X(j) for these {i,j}: {512, 5969}, {669, 5106}, {3124, 11182}, {5970, 99}

X(14607) = X(2)X(6)∩X(99)X(512)

Barycentrics    (a - b)*(a + b)*(a - c)*(a + c)*(a^4*b^2 - 2*a^2*b^4 + a^4*c^2 + b^4*c^2 - 2*a^2*c^4 + b^2*c^4) : :

X(14607) lies on the cubic K017 and these lines: {2, 6}, {99, 512}, {110, 4590}, {523, 4576}, {669, 1634}, {670, 850}, {5182, 5970}

X(14607) = reflection of X(385) in X(3231)
X(14607) = isogonal conjugate of X(14606)
X(14607) = X(11182)-cross conjugate of X(5969)
X(14607) = X(6)-Hirst inverse of X(5468)
X(14607) = X(2)-line conjugate of X(2086)
X(14607) = X(i)-vertex conjugate of X(j) for these (i,j): {669, 5468}, {5468, 669}
X(14607) = cevapoint of X(5969) and X(11182)
X(14607) = trilinear pole of line X(5106)X(5969)
X(14607) = crossdifference of every pair of points on line {512, 2086}
X(14607) = X(661)-isoconjugate of X(5970)
X(14607) = barycentric product X(i)*X(j) for these {i,j}: {99, 5969}, {670, 5106}, {4590, 11182}
X(14607) = barycentric quotient X(i)/X(j) for these {i,j}: {110, 5970}, {5106, 512}, {5969, 523}, {11182, 115}

X(14608) =  X(2)X(512)∩X(6)X(99)

Barycentrics    (-2*a^2 + b^2 + c^2)*(a^2*b^2 - 2*a^2*c^2 + b^2*c^2)*(-2*a^2*b^2 + a^2*c^2 + b^2*c^2) : :

X(14608) lies on the cubics K017, K280, K553, and on these lines: {2, 512}, {6, 99}, {76, 886}, {187, 5468}, {249, 1501}, {843, 5108}, {5118, 13586}

X(14608) = isogonal conjugate of X(14609)
X(14608) = trilinear pole of line X(351)X(524)
X(14608) = crossdifference of every pair of points on line {888, 3231}
X(14608) = X(i)-isoconjugate of X(j) for these (i,j): {111, 2234}, {538, 923}, {897, 3231}
X(14608) = barycentric product X(i)*X(j) for these {i,j}: {351, 886}, {524, 3228}, {690, 9150}, {729, 3266}
X(14608) = barycentric quotient X(i)/X(j) for these {i,j}: {187, 3231}, {351, 888}, {524, 538}, {690, 9148}, {729, 111}, {896, 2234}, {3228, 671}, {5467, 5118}, {9150, 892}, {9155, 6786}

X(14609) = X(2)X(99)∩X(6)X(512)

Barycentrics    a^2*(a^2 + b^2 - 2*c^2)*(a^2 - 2*b^2 + c^2)*(a^2*b^2 + a^2*c^2 - 2*b^2*c^2) : :

X(14609) lies on the cubics K017 and K281, and on these lines: {2, 99}, {6, 512}, {32, 691}, {39, 5968}, {187, 11634}, {892, 7798}, {895, 5028}, {3231, 5118}, {3291, 9177}, {7739, 9214}, {7772, 14246}

X(14609) = isogonal conjugate of X(14608)
X(14609) = crossdifference of every pair of points on line {351, 524}
X(14609) = trilinear pole of line X(888)X(3231)
X(14609) = X(i)-isoconjugate of X(j) for these (i,j): {729, 14210}, {896, 3228}, {2642, 9150}
X(14609) = barycentric product X(i)*X(j) for these {i,j}: {111, 538}, {671, 3231}, {691, 9148}, {888, 892}, {897, 2234}, {5118, 5466}, {6786, 9154}
X(14609) = barycentric quotient X(i)/X(j) for these {i,j}: {111, 3228}, {538, 3266}, {691, 9150}, {887, 351}, {888, 690}, {892, 886}, {2234, 14210}, {3231, 524}, {5118, 5468}

X(14610) =  MIDPOINT OF X(110) AND X(13290)

Barycentrics    (b - c)*(b + c)*(5*a^4 - 3*a^2*b^2 + b^4 - 3*a^2*c^2 - b^2*c^2 + c^4) : :
X(14610) = X(351) - 3 X(9123) = 3 X(9123) + X(9131) = 3 X(9125) - X(9134) = 5 X(351) - 3 X(9185) = 5 X(9123) - X(9185) = 5 X(9131) + 3 X(9185) = 9 X(9185) - 5 X(9979) = 3 X(351) - X(9979) = 9 X(9123) - X(9979) = 3 X(9131) + X(9979) = 3 X(10190) - 2 X(14417) = 3 X(9168) - X(14424)

X(14610) lies on these lines: {110, 930}, {351, 523}, {525, 5027}, {804, 10190}, {900, 9810}, {1649, 8786}, {3566, 9135}, {3800, 9208}, {4977, 9811}, {5113, 7927}, {7950, 14316}, {9007, 13302}, {9125, 9134}, {9147, 9479}, {9168, 14424}, {10278, 11176}

X(14610) = midpoint of X(i) and X(j) for these {i,j}: {110, 13290}, {351, 9131}, {1649, 9485}, {9147, 11123}
X(14610) = reflection of X(10278) in X(11176)
X(14610) = crosspoint of X(99) and X(9227)
X(14610) = crosssum of X(512) and X(9225)
X(14610) = crossdifference of every pair of points on line {574, 3981}
X(14610) = X(5)-of-1st-Parry-triangle
X(14610) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (9123, 9131, 351), (13316, 13317, 9135)

X(14611) =  MIDPOINT OF X(110) AND X(14480)

Barycentrics    (a - b)*(a + b)*(a - c)*(a + c)*(2*a^8 - 2*a^6*b^2 - 3*a^4*b^4 + 4*a^2*b^6 - b^8 - 2*a^6*c^2 + 8*a^4*b^2*c^2 - 4*a^2*b^4*c^2 + 4*b^6*c^2 - 3*a^4*c^4 - 4*a^2*b^2*c^4 - 6*b^4*c^4 + 4*a^2*c^6 + 4*b^2*c^6 - c^8) : :
X(14611) = 3 X(110) - X(476) = 3 X(110) - 2 X(3233) = 2 X(476) - 3 X(7471) = 4 X(3233) - 3 X(7471) = X(7471) + 2 X(14480) = X(476) + 3 X(14480) = 2 X(3233) + 3 X(14480)
X

X(14611) lies on these lines: {2, 9717}, {30, 146}, {99, 6563}, {110, 476}, {147, 858}, {195, 10223}, {385, 7426}, {477, 14094}, {526, 11751}, {542, 3258}, {648, 1302}, {1483, 3109}, {1553, 6053}, {1995, 2452}, {3154, 3448}, {5972, 6070}, {9137, 9182}, {11101, 13869}

X(14611) = midpoint of X(i) and X(j) for these {i,j}: {110, 14480}, {477, 14094}, X(14611) = reflection of X(i) in X(j) for these {i,j}: {476, 3233}, {1553, 6053}, {3448, 3154}, {6070, 5972}, {7471, 110}
X(14611) = anticomplement X(12079)
X(14611) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {1101, 30}, {9274, 6740}
X(14611) = trilinear pole of line X(6128)X(6699)
X(14611) = barycentric product X(i)*X(j) for these {i,j}: {99, 6128}, {648, 6699}
X(14611) = barycentric quotient X(i)/X(j) for these {i,j}: {6128, 523}, {6699, 525}
X(14611) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (110, 476, 3233), (476, 3233, 7471)

X(14612) = X(56)X(7760)∩X(99)X(1415)

Barycentrics    (a - b)*(a - c)*(a + b - c)*(a - b + c)*(a^3 + a*b^2 + a*b*c + b^2*c + a*c^2 + b*c^2) : :

X(14612) lies on these lines: {56, 7760}, {99, 1415}, {101, 514}, {108, 648}, {109, 190}, {645, 651}, {671, 13273}, {1025, 6649}, {1470, 7757}, {3570, 4551}

X(14612) = barycentric product X(i)*X(j) for these {i,j}: {664, 5263}, {4554, 5276}
X(14612) = barycentric quotient X(i)/X(j) for these {i,j}: {5263, 522}, {5276, 650}

X(14613) =  X(101)X(668)∩X(110)X(670)

Barycentrics    b*(-a + b)*c*(-a + c)*(-a^2 + b*c)*(a^4 + a^3*b + a^2*b^2 + a^3*c + 2*a^2*b*c + a*b^2*c + a^2*c^2 + a*b*c^2 + b^2*c^2) : :

X(14613) lies on these lines: {101, 668}, {110, 670}, {692, 1978}

X(14614) =  X(2)X(6)∩X(3)X(6179)

Barycentrics    3*a^4 + a^2*b^2 + a^2*c^2 - 2*b^2*c^2 : :
X(14613) = 4 X(32) - X(1975) = X(315) - 4 X(5305) = 2 X(626) - 5 X(5346) = 2 X(32) + X(7754) = X(1975) + 2 X(7754) = 4 X(5306) - X(7788) = X(7754) - 4 X(7805) = X(32) + 2 X(7805) = X(1003) + 4 X(7805) = X(1975) + 8 X(7805) = 5 X(1975) - 8 X(7816) = 5 X(1003) - 4 X(7816) = 5 X(32) - 2 X(7816) = 5 X(7805) + X(7816) = 5 X(7754) + 4 X(7816) = 2 X(315) - 5 X(7851) = 8 X(5305) - 5 X(7851) = 4 X(6680) - X(7855) = 8 X(6680) - 5 X(7881) = 2 X(7855) - 5 X(7881) = 5 X(7867) - 2 X(7882) = 4 X(7880) - 5 X(8366) = X(5989) - 4 X(12829)

X(14614) lies on these lines:
{2, 6}, {3, 6179}, {22, 5201}, {25, 648}, {32, 538}, {76, 11286}, {98, 1351}, {99, 1384}, {187, 7798}, {190, 3052}, {194, 3053}, {251, 9462}, {262, 5093}, {315, 5305}, {381, 7812}, {511, 9755}, {576, 13860}, {598, 12156}, {626, 5346}, {671, 3830}, {754, 5309}, {1078, 7894}, {1353, 9744}, {1383, 9870}, {1447, 3759}, {1627, 4558}, {1656, 7858}, {2549, 8353}, {2871, 3060}, {2896, 7920}, {3114, 11335}, {3225, 3511}, {3534, 9301}, {3564, 9753}, {3734, 5008}, {3758, 7081}, {3767, 7762}, {3788, 7890}, {3793, 8354}, {3849, 11648}, {3933, 8368}, {4396, 5332}, {4400, 7296}, {4428, 4664}, {5007, 7751}, {5013, 7793}, {5017, 5969}, {5023, 7783}, {5024, 7771}, {5041, 7815}, {5077, 11057}, {5085, 6194}, {5102, 9756}, {5182, 5976}, {5286, 7750}, {5319, 6656}, {5355, 7761}, {5368, 7826}, {5939, 10754}, {5980, 11485}, {5981, 11486}, {5999, 11477}, {6680, 7855}, {7739, 8356}, {7746, 7838}, {7755, 7759}, {7758, 7807}, {7767, 7803}, {7768, 7856}, {7772, 7780}, {7776, 7828}, {7784, 7797}, {7785, 13881}, {7799, 11288}, {7809, 11318}, {7811, 7827}, {7817, 7818}, {7829, 7854}, {7844, 7845}, {7848, 7913}, {7850, 7919}, {7852, 7896}, {7857, 7905}, {7867, 7882}, {7873, 7902}, {7874, 7916}, {7880, 8366}, {7883, 7884}, {7886, 7903}, {7899, 7949}, {7901, 7946}, {7907, 13571}, {7917, 7942}, {7923, 7929}, {7924, 9939}, {7926, 14061}, {7932, 7939}, {8267, 14570}, {9307, 9909}, {9308, 10311}, {10033, 14492}, {10845, 11917}, {10846, 11916}, {11054, 11159}, {11317, 14537}, {13111, 14269}

X(14613) = midpoint of X(1003) and X(7754)
X(14613) = reflection of X(i) in X(j) for these {i,j}: {2, 5306}, {1003, 32}, {1975, 1003}, {3933, 8368}, {7788, 2}, {7818, 7817}, {7841, 5309}
X(14613) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 385, 8667), (2, 7837, 9766), (2, 8667, 183), (6, 183, 11174), (6, 385, 183), (6, 8667, 2), (32, 7754, 1975), (32, 7805, 7754), (69, 5304, 7792), (69, 7792, 7868), (76, 12150, 11286), (193, 7735, 325), (194, 13586, 8716), (230, 3629, 7774), (315, 5305, 7851), (385, 7766, 6), (1078, 7894, 9605), (3053, 8716, 13586), (3767, 7762, 7773), (5007, 7751, 7770), (5319, 14023, 6656), (5368, 7826, 7834), (6144, 7778, 7779), (6179, 7760, 3), (6680, 7855, 7881), (7755, 7759, 7887), (7768, 7856, 7866), (7772, 7780, 11285), (7779, 7806, 7778), (7793, 7839, 5013), (7797, 7893, 7784), (7811, 7827, 11287), (7812, 14568, 381), (7826, 7834, 7879), (7828, 7877, 7776), (8584, 13468, 9300), (9300, 13468, 2)

X(14615) = ISOTOMIC CONJUGATE OF X(64)

Barycentrics    b^2*c^2*(-3*a^4 + 2*a^2*b^2 + b^4 + 2*a^2*c^2 - 2*b^2*c^2 + c^4) : :
Barycentrics    cot A - csc A cos B cos C : :
Barycentrics    (csc^2 A)(tan B + tan C - tan A) : :
Barycentrics    (S^2 - 2 SB SC)/(SB + SC) : :

Let A'B'C' be the midheight triangle. (The lines AA', BB', CC' concur in X(4).) Let A" be the trilinear pole of line B'C', and define B" and C" cyclically. (The lines AA", BB", CC" concur in X(20).) Let A* be the trilinear pole of line B"C", and define B* and C* cyclically. (The lines AA*, BB*, CC* concur in X(69).) Let A** be the trilinear pole of line B*C*, and define B** and C** cyclically. The lines AA**, BB**, CC** concur in X(14615). (Randy Hutson, November 2, 2017)

X(14615) lies on the cubics K183 and K235 and on these lines: {2, 800}, {4, 69}, {75, 1088}, {95, 7771}, {99, 1294}, {183, 5020}, {253, 305}, {290, 6391}, {304, 309}, {325, 1368}, {394, 801}, {668, 1264}, {1272, 6188}, {3346, 3926}, {3718, 8806}, {7767, 9825}, {7782, 9723}, {8024, 10513}

X(14615) = isogonal conjugate of X(33581)
X(14615) = isotomic conjugate of X(64)
X(14615) = anticomplement X(800)
X(14615) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {775, 2}, {801, 8}, {821, 6515}, {1105, 5905}
X(14615) = X(305)-Ceva conjugate of X(76)
X(14615) = cevapoint of X(i) and X(j) for these (i,j): {2, 6225}, {69, 6527}, {394, 11413}
X(14615) = X(2883)-cross conjugate of X(2)
X(14615) = trilinear pole of line X(6587)X(20580)
X(14615) = X(7257)-beth conjugate of X(1264)
X(14615) = polar conjugate of isogonal conjugate of X(37669)
X(14615) = X(i)-isoconjugate of X(j) for these (i,j): {6, 2155}, {31, 64}, {32, 2184}, {253, 560}, {459, 9247}, {810, 1301}, {1073, 1973}, {1096, 14379}, {2175, 8809}
X(14615) = barycentric product X(i)*X(j) for these {i,j}: {20, 76}, {154, 1502}, {304, 1895}, {305, 1249}, {310, 8804}, {561, 610}, {670, 6587}, {3198, 6385}, {3926, 14249}, {4572, 14331}, {6331, 8057}
X(14615) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 2155}, {2, 64}, {20, 6}, {69, 1073}, {75, 2184}, {76, 253}, {85, 8809}, {122, 3269}, {154, 32}, {204, 1973}, {264, 459}, {311, 13157}, {343, 8798}, {394, 14379}, {610, 31}, {648, 1301}, {1097, 610}, {1249, 25}, {1394, 604}, {1895, 19}, {2052, 6526}, {2883, 800}, {3079, 3172}, {3172, 1974}, {3198, 213}, {3213, 1395}, {5930, 1400}, {6525, 2207}, {6527, 3343}, {6587, 512}, {6616, 1033}, {7070, 41}, {7156, 2212}, {8057, 647}, {8804, 42}, {10152, 8749}, {11064, 11589}, {11413, 14390}, {14249, 393}, {14308, 3709}, {14331, 663}, {14345, 9409}
X(14615) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (69, 264, 76), (69, 3260, 264), (309, 322, 304)

X(14616) =  ISOGONAL CONJUGATE OF X(3724)

Barycentrics    b*(a + b)*c*(a + c)*(a^2 - a*b + b^2 - c^2)*(-a^2 + b^2 + a*c - c^2) : :

X(14616) lies on the Steiner circumellipse and these lines: {75, 99}, {80, 313}, {86, 664}, {92, 648}, {94, 1029}, {190, 321}, {265, 8044}, {274, 4597}, {286, 7282}, {561, 670}, {662, 4858}, {666, 2341}, {799, 1227}, {1168, 4555}, {1821, 2966}, {2185, 14213}, {3112, 4577}, {6648, 14534}

X(14616) = isogonal conjugate of X(3724)
X(14616) = isotomic conjugate of X(758)
X(14616) = X(i)-cross conjugate of X(j) for these (i,j): {758, 2}, {3738, 662}, {3762, 799}, {5080, 14534}, {14206, 85}, {14304, 823}
X(14616) = cevapoint of X(i) and X(j) for these (i,j): {2, 758}, {81, 1325}, {3738, 4858}, {6370, 8287}
X(14616) = trilinear pole of line X(2)X(1577)
X(14616) = X(314)-beth conjugate of X(99)
X(14616) = X(i)-zayin conjugate of X(j) for these (i,j): {1, 3724}, {43, 2245}
X(14616) = X(i)-isoconjugate of X(j) for these (i,j): {1, 3724}, {6, 2245}, {31, 758}, {32, 3936}, {36, 42}, {37, 7113}, {50, 8818}, {55, 1464}, {65, 2361}, {163, 2610}, {184, 860}, {212, 1835}, {213, 3218}, {228, 1870}, {320, 1918}, {654, 4559}, {661, 1983}, {798, 4585}, {810, 4242}, {1333, 4053}, {1400, 2323}, {1402, 4511}, {1576, 6370}, {2171, 4282}, {4551, 8648}, {6742, 14270}
X(14616) = barycentric product X(i)*X(j) for these {i,j}: {76, 759}, {80, 274}, {85, 6740}, {310, 2161}, {314, 2006}, {331, 1793}, {2341, 6063}, {6187, 6385}
X(14616) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 2245}, {2, 758}, {6, 3724}, {10, 4053}, {21, 2323}, {27, 1870}, {57, 1464}, {58, 7113}, {60, 4282}, {75, 3936}, {80, 37}, {81, 36}, {86, 3218}, {92, 860}, {94, 6757}, {99, 4585}, {110, 1983}, {274, 320}, {278, 1835}, {284, 2361}, {333, 4511}, {523, 2610}, {648, 4242}, {655, 4551}, {693, 4707}, {759, 6}, {1411, 1400}, {1434, 1443}, {1577, 6370}, {1793, 219}, {1807, 71}, {2006, 65}, {2161, 42}, {2166, 8818}, {2222, 4559}, {2341, 55}, {2605, 2624}, {2611, 2088}, {3737, 654}, {3936, 4736}, {4282, 215}, {4560, 3738}, {5235, 4867}, {5333, 4880}, {6187, 213}, {6740, 9}, {7192, 3960}, {7199, 4453}, {7252, 8648}, {8025, 4973}, {9273, 1101}, {14206, 6739}

X(14617) =  X(2)X(1501)∩X(6)X(706)

Barycentrics    (a^2 - a*b + b^2)*(a^2 + a*b + b^2)*(b^2 + c^2)*(a^2 - a*c + c^2)*(a^2 + a*c + c^2) : :

X(14617) lies on the cubic K421 and these lines: {2, 1501}, {6, 706}, {141, 8623}, {384, 694}, {512, 7804}, {703, 9063}, {732, 3051}, {1613, 11324}, {2295, 3113}, {3117, 4048}, {3329, 8842}, {3589, 3613}, {10000, 11196}, {10007, 14096}, {10337, 10341}

X(14617) = cevapoint of X(i) and X(j) for these (i,j): {384, 3329}, {3051, 14096}
X(14617) = crosspoint of X(3114) and X(3407)
X(14617) = trilinear pole of line X(826)X(9494)
X(14617) = crosssum of X(3094) and X(3117)
X(14617) = X(3407)-daleth conjugate of X(8840)
X(14617) = X(14617) = X(i)-isoconjugate of X(j) for these (i,j): {82, 3094}, {83, 3116}, {3112, 3117}
X(14617) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 3407, 8840)
X(14617) = barycentric product X(i)*X(j) for these {i,j}: {38, 3113}, {39, 3114}, {141, 3407}, {688, 9063}
X(14617) = barycentric quotient X(i)/X(j) for these {i,j}: {39, 3094}, {141, 3314}, {427, 5117}, {732, 9865}, {1964, 3116}, {3051, 3117}, {3113, 3112}, {3114, 308}, {3407, 83}, {9494, 9006}

X(14618) =  ISOTOMIC CONJUGATE OF X(4558)

Barycentrics    b^2*(b - c)*c^2*(b + c)*(-a^2 + b^2 - c^2)*(a^2 + b^2 - c^2) : :
Barycentrics    sec A sin(B - C) : :

X(14618) lies on these lines: {4, 512}, {25, 4108}, {107, 935}, {112, 2966}, {264, 8430}, {297, 525}, {324, 14592}, {403, 523}, {427, 5996}, {520, 6761}, {647, 14165}, {687, 4558}, {924, 5962}, {1019, 5307}, {1826, 4129}, {1869, 4807}, {2052, 2394}, {2489, 4580}, {2506, 5286}, {2799, 3267}, {2970, 10555}, {4064, 4086}, {5489, 13450}, {7648, 7803}

X(14618) = isogonal conjugate of X(32661)
X(14618) = isotomic conjugate of X(4558)
X(14618) = polar conjugate of X(110)
X(14618) = X(92)-beth conjugate of X(7178)
X(14618) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {91, 13219}, {925, 4329}
X(14618) = X(i)-Ceva conjugate of X(j) for these (i,j): {264, 2970}, {648, 324}, {1969, 2973}, {2052, 338}, {6331, 264}, {6528, 4}
X(14618) = X(i)-cross conjugate of X(j) for these (i,j): {115, 4}, {338, 2052}, {523, 850}, {1637, 10412}, {2970, 264}, {12077, 523}
X(14618) = pole wrt polar circle of Brocard axis
X(14618) = X(i)-isoconjugate of X(j) for these (i,j): {3, 163}, {6, 4575}, {31, 4558}, {32, 4592}, {48, 110}, {58, 906}, {63, 1576}, {99, 9247}, {101, 1437}, {107, 4100}, {109, 2193}, {112, 255}, {162, 577}, {184, 662}, {212, 4565}, {228, 4556}, {249, 810}, {250, 822}, {283, 1415}, {304, 14574}, {560, 4563}, {563, 925}, {603, 5546}, {647, 1101}, {692, 1790}, {799, 14575}, {811, 14585}, {827, 4020}, {849, 4574}, {1110, 7254}, {1331, 1333}, {1332, 2206}, {1408, 4587}, {1409, 4636}, {1625, 2169}, {1813, 2194}, {2204, 6517}, {2315, 10420}, {2617, 14533}
X(14618) = cevapoint of X(i) and X(j) for these (i,j): {6, 13558}, {523, 2501}, {647, 924}
X(14618) = cevapoint of polar circle intercepts of Brocard axis
X(14618) = crosspoint of X(i) and X(j) for these (i,j): {264, 6331}, {275, 648}
X(14618) = trilinear pole of line X(125)X(136)
X(14618) = crossdifference of every pair of points on line {184, 418}
X(14618) = crosssum of X(i) and X(j) for these (i,j): {184, 3049}, {216, 647}, {520, 3917}
X(14618) = barycentric product X(i)*X(j) for these {i,j}: {4, 850}, {76, 2501}, {92, 1577}, {99, 2970}, {107, 339}, {115, 6331}, {125, 6528}, {158, 14208}, {264, 523}, {273, 4086}, {276, 12077}, {286, 4036}, {313, 7649}, {318, 4077}, {331, 3700}, {338, 648}, {340, 10412}, {349, 3064}, {393, 3267}, {525, 2052}, {661, 1969}, {670, 8754}, {683, 12075}, {811, 1109}, {847, 6563}, {1093, 3265}, {1502, 2489}, {1826, 3261}, {2592, 2593}, {2971, 4609}, {2973, 3952}, {3268, 6344}, {6368, 8795}, {7017, 7178}, {7141, 7192}, {14165, 14592}
X(14618) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 4575}, {2, 4558}, {4, 110}, {10, 1331}, {19, 163}, {25, 1576}, {27, 4556}, {29, 4636}, {37, 906}, {53, 1625}, {75, 4592}, {76, 4563}, {92, 662}, {93, 930}, {107, 250}, {115, 647}, {125, 520}, {136, 924}, {158, 162}, {162, 1101}, {225, 109}, {226, 1813}, {235, 1624}, {254, 13398}, {264, 99}, {273, 1414}, {278, 4565}, {281, 5546}, {297, 2421}, {307, 6517}, {313, 4561}, {318, 643}, {321, 1332}, {324, 14570}, {331, 4573}, {338, 525}, {339, 3265}, {340, 10411}, {393, 112}, {427, 1634}, {468, 5467}, {512, 184}, {513, 1437}, {514, 1790}, {520, 1092}, {522, 283}, {523, 3}, {525, 394}, {594, 4574}, {647, 577}, {648, 249}, {650, 2193}, {656, 255}, {661, 48}, {669, 14575}, {690, 3292}, {693, 1444}, {798, 9247}, {822, 4100}, {826, 3917}, {847, 925}, {850, 69}, {868, 684}, {924, 1147}, {933, 14587}, {1086, 7254}, {1093, 107}, {1109, 656}, {1235, 4576}, {1300, 10420}, {1441, 6516}, {1510, 49}, {1577, 63}, {1637, 3284}, {1824, 692}, {1826, 101}, {1847, 4637}, {1880, 1415}, {1897, 4570}, {1969, 799}, {1974, 14574}, {1990, 2420}, {2052, 648}, {2321, 4587}, {2395, 248}, {2422, 14600}, {2485, 10316}, {2489, 32}, {2492, 10317}, {2501, 6}, {2533, 3955}, {2588, 1823}, {2589, 1822}, {2592, 8116}, {2593, 8115}, {2616, 2169}, {2623, 14533}, {2643, 810}, {2969, 3733}, {2970, 523}, {2971, 669}, {2973, 7192}, {3049, 14585}, {3064, 284}, {3120, 1459}, {3124, 3049}, {3239, 2327}, {3265, 3964}, {3267, 3926}, {3566, 3167}, {3569, 3289}, {3700, 219}, {3701, 4571}, {3708, 822}, {3800, 3796}, {3801, 3784}, {4010, 7193}, {4017, 603}, {4024, 71}, {4036, 72}, {4041, 212}, {4049, 1797}, {4064, 3682}, {4077, 77}, {4079, 2200}, {4086, 78}, {4088, 1818}, {4122, 3781}, {4171, 1802}, {4391, 1812}, {4397, 1792}, {4404, 4855}, {4466, 4091}, {4516, 1946}, {4581, 1798}, {4705, 228}, {4815, 4652}, {5094, 9145}, {5139, 8651}, {5466, 895}, {5489, 2972}, {6331, 4590}, {6332, 6514}, {6335, 4567}, {6336, 4591}, {6344, 476}, {6368, 5562}, {6521, 823}, {6526, 1301}, {6530, 4230}, {6531, 2715}, {6563, 9723}, {6591, 1333}, {6753, 571}, {7017, 645}, {7101, 7259}, {7140, 4557}, {7141, 3952}, {7178, 222}, {7216, 7099}, {7649, 58}, {8058, 1819}, {8061, 4020}, {8611, 2289}, {8735, 7252}, {8736, 4559}, {8737, 5995}, {8738, 5994}, {8754, 512}, {8801, 907}, {8882, 14586}, {8884, 933}, {9134, 8681}, {10151, 5502}, {10412, 265}, {12075, 6467}, {12077, 216}, {12079, 14380}, {13400, 8573}, {14165, 14590}, {14208, 326}, {14273, 187}, {14295, 12215}

X(14619) =  X(31)X(39)∩X(75)X(83)

Barycentrics    a^2*(a^2 + a*b + b^2 + c^2)*(a^2 + b^2 + a*c + c^2)*(a^2*b^2 + a^2*b*c + a^2*c^2 + b^2*c^2) : :

X(14619) lies on the cubic K507 and these lines: {31, 39}, {75, 83}

X(14620) =  X(10)X(75)∩X(31)X(32)

Barycentrics    a^3*(b^4 + b^3*c + b^2*c^2 + b*c^3 + c^4) : :

X(14620) lies on the cubic K507 and lines: {10, 75}, {31, 32}, {39, 3116}, {100, 735}, {101, 747}, {2085, 2276}, {4116, 4531}

X(14620) = crossdifference of every pair of points on line {693, 1919}
X(14620) = X(667)-isoconjugate of X(9065)
X(14620) = barycentric product X(1978)X(9008)
X(14620) = barycentric quotient X(i)/X(j) for these {i,j}: {190, 9065}, {9008, 649}

X(14621) =  ISOGONAL CONJUGATE OF X(2276)

Barycentrics   (a^2 + a*b + b^2)*(a^2 + a*c + c^2) : :

X(14621) lies on the circumconic {A,B,C,X(2),X(7)}, the cubics K421 and K507, and one thesse lines:
{1, 335}, {2, 31}, {6, 75}, {7, 604}, {27, 2203}, {57, 7249}, {81, 310}, {86, 1333}, {226, 3407}, {273, 608}, {673, 1492}, {675, 825}, {739, 789}, {903, 4586}, {940, 2162}, {1088, 1407}, {1240, 2298}, {2295, 3113}, {3416, 3661}, {3797, 3923}, {4393, 4649}, {4676, 6651}, {4817, 6548}, {5381, 5388}, {5711, 7770}, {5749, 5936}

X(14621) = isogonal conjugate of X(2276)
X(14621) = isotomic conjugate of X(3661)
X(14621) = X(i)-Ceva conjugate of X(j) for these (i,j): {4586, 4817}, {5388, 789}
X(14621) = X(i)-cross conjugate of X(j) for these (i,j): {3821, 75}, {4784, 99}, {4785, 190}, {4817, 4586}
X(14621) = cevapoint of X(i) and X(j) for these (i,j): {2, 4393}, {6, 1001}, {985, 2344}
X(14621) = crosspoint of X(789) and X(5338)
X(14621) = trilinear pole of line X(514)X(659)
X(14621) = crossdifference of every pair of points on line {788, 3250}
X(14621) = complementary conjugate of anticomplement of X(21264)
X(14621) = X(2344)-beth conjugate of X(6)
X(14621) = X(i)-zayin conjugate of X(j) for these (i,j): {1, 2276}, {4040, 1491}
X(14621) = X(i)-isoconjugate of X(j) for these (i,j): {1, 2276}, {2, 869}, {6, 984}, {9, 1469}, {19, 3781}, {31, 3661}, {37, 3736}, {41, 7179}, {55, 7146}, {57, 4517}, {86, 3774}, {100, 3250}, {101, 1491}, {163, 4122}, {190, 788}, {220, 7204}, {238, 3862}, {292, 3783}, {604, 3790}, {649, 3799}, {667, 3807}, {692, 824}, {983, 3094}, {1252, 4475}, {1333, 3773}, {1400, 3786}, {1415, 4522}, {1911, 3797}, {1914, 3864}, {1919, 4505}, {1978, 8630}, {2161, 3792}, {2279, 3789}, {2344, 12837}, {3117, 7033}, {4439, 9456}, {4481, 4557}, {4555, 14436}
X(14621) = barycentric product X(i)*X(j) for these {i,j}: {1, 870}, {31, 871}, {75, 985}, {85, 2344}, {190, 4817}, {513, 789}, {514, 4586}, {693, 1492}, {825, 3261}, {982, 3113}, {1015, 5388}, {1111, 5384}, {2275, 3114}, {3407, 3662}, {4613, 7192}
X(14621) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 984}, {2, 3661}, {3, 3781}, {6, 2276}, {7, 7179}, {8, 3790}, {10, 3773}, {21, 3786}, {31, 869}, {36, 3792}, {55, 4517}, {56, 1469}, {57, 7146}, {58, 3736}, {100, 3799}, {190, 3807}, {213, 3774}, {238, 3783}, {239, 3797}, {244, 4475}, {269, 7204}, {291, 3864}, {292, 3862}, {513, 1491}, {514, 824}, {519, 4439}, {522, 4522}, {523, 4122}, {551, 4407}, {649, 3250}, {667, 788}, {668, 4505}, {789, 668}, {812, 4486}, {825, 101}, {870, 75}, {871, 561}, {985, 1}, {1001, 3789}, {1019, 4481}, {1125, 3775}, {1469, 12837}, {1492, 100}, {1980, 8630}, {2275, 3094}, {2344, 9}, {3113, 7033}, {3662, 3314}, {3736, 4476}, {4367, 3805}, {4586, 190}, {4613, 3952}, {4777, 4951}, {4778, 4818}, {4817, 514}, {5384, 765}, {7032, 3116}, {8300, 3802}

X(14622) = X(31)X(83)∩X(39)X(75)

Barycentrics    (a^2 + b^2 + b*c + c^2)*(a^2*b^2 + a*b^2*c + a^2*c^2 + b^2*c^2)*(a^2*b^2 + a^2*c^2 + a*b*c^2 + b^2*c^2) : :

X(14622) lies on the cubic K507 and these lines: {31, 83}, {39, 75}

X(14623) = X(31)X(76)∩X(32)X(75)

Barycentrics    b*(a^4 + a^3*b + a^2*b^2 + a*b^3 + b^4)*c*(a^4 + a^3*c + a^2*c^2 + a*c^3 + c^4) : :

X(14623) lies on the cubic K507 and these lines: {31, 76}, {32, 75}, {58, 6385}, {85, 1397}, {274, 2206}, {331, 1395}, {334, 1922}, {727, 9065}, {6383, 7121}

X(14623) = X(668)-isoconjugate of X(9008)
X(14623) = trilinear pole of line X(692)X(1919)
X(14623) = barycentric product X(649)X(9065)
X(14623) = barycentric quotient X(i)/X(j) for these {i,j}: {1919, 9008}, {9065, 1978}

X(14624) =  X(1)X(10469)∩X(6)X(8)

Barycentrics    (b + c)*(a^2 + b^2 + a*c + b*c)*(a^2 + a*b + b*c + c^2) : :

X(14624) lies on the circumconic {A,B,C,X(2),X(6)}, the cubic K696, and these lines: {1, 10469}, {2, 1240}, {6, 8}, {10, 1400}, {25, 281}, {37, 3701}, {42, 2321}, {111, 8707}, {346, 941}, {956, 961}, {967, 1150}, {1018, 2269}, {1169, 5291}, {1171, 6539}, {1215, 2171}, {1427, 1441}, {2359, 10570}, {3572, 4581}, {4037, 9281}, {5435, 5936}, {5750, 10459}

X(14624) = isotomic conjugate of X(16705)
X(14624) = crosssum of X(1193) and X(2300)
X(14624) = cevapoint of X(i) and X(j) for these (i,j): {10, 1215}, {37, 594}, {210, 1500}
X(14624) = X(i)-cross conjugate of X(j) for these (i,j): {37, 2298}, {650, 1018}, {4140, 4552}, {4705, 3952}
X(14624) = X(i)-isoconjugate of X(j) for these (i,j): {57, 4267}, {58, 3666}, {81, 1193}, {86, 2300}, {163, 3004}, {593, 2292}, {662, 6371}, {757, 2092}, {849, 1211}, {960, 1412}, {1014, 2269}, {1333, 4357}, {1408, 3687}, {1437, 1848}, {1444, 2354}, {1509, 3725}, {1576, 4509}, {1790, 1829}, {2194, 3674}, {3733, 3882}
X(14624) = trilinear pole of line {512, 3700}
X(14624) = barycentric product X(i)*X(j) for these {i,j}: {10, 1220}, {42, 1240}, {321, 2298}, {523, 8707}, {594, 14534}, {961, 3701}, {1089, 2363}, {1798, 7141}, {3700, 6648}, {3952, 4581}
X(14624) = barycentric quotient X(i)/X(j) for these {i,j}: {10, 4357}, {37, 3666}, {42, 1193}, {55, 4267}, {210, 960}, {213, 2300}, {226, 3674}, {512, 6371}, {523, 3004}, {594, 1211}, {756, 2292}, {872, 3725}, {961, 1014}, {1018, 3882}, {1169, 593}, {1220, 86}, {1240, 310}, {1334, 2269}, {1500, 2092}, {1577, 4509}, {1791, 1444}, {1824, 1829}, {1826, 1848}, {2298, 81}, {2321, 3687}, {2333, 2354}, {2359, 1790}, {2363, 757}, {3700, 3910}, {4515, 3965}, {4581, 7192}, {6057, 3704}, {6648, 4573}, {7140, 429}, {8687, 4565}, {8707, 99}, {14534, 1509}

X(14625) =  X(6)X(7)∩X(10)X(1018)

Barycentrics    (b + c)*(3*a + b + c)*(a^2 + b^2 - a*c - b*c)*(-a^2 + a*b + b*c - c^2) : :

X(14625) lies on the cubic K696 and these lines: {6, 7}, {10, 1018}, {3616, 4258}

X(14625) = cevapoint of X(4771) and X(5257)
X(14625) = trilinear pole of line X(1481(X(8653)
X(14625) = X(940)-zayin conjugate of X(672)
X(14625) = X(i)-isoconjugate of X(j) for these (i,j): {665, 4614}, {2254, 4627}
X(14625) = barycentric product X(i)*X(j) for these {i,j}: {666, 4841}, {673, 5257}, {927, 4843}, {3616, 13576}
X(14625) = barycentric quotient X(i)/X(j) for these {i,j}: {666, 4633}, {919, 4627}, {3671, 9436}, {4061, 3717}, {4822, 2254}, {4832, 665}, {4841, 918}, {5257, 3912}, {8653, 926}, {13576, 5936}

X(14626) =  X(6)X(1334)∩X(7)X(10)

Barycentrics    a^2*(a + 3*b + c)*(a + b + 3*c)*(a*b - b^2 + a*c - c^2) : :

X(14626) lies on the cubic K696 and these lines: {6, 1334}, {7, 10}, {513, 4041}, {840, 8694}, {1002, 3720}, {2340, 3286}

X(14626) = crossdifference of every pair of points on line {1449, 4778}
X(14626) = X(i)-isoconjugate of X(j) for these (i,j): {105, 3616}, {391, 1462}, {666, 4790}, {673, 1449}, {919, 4801}, {1416, 4673}
X(14626) = barycentric product X(i)*X(j) for these {i,j}: {241, 4866}, {672, 5936}, {918, 8694}, {926, 4624}, {2254, 4606}, {2334, 3912}, {4088, 4627}
X(14626) = barycentric quotient X(i)/X(j) for these {i,j}: {665, 4778}, {672, 3616}, {926, 4765}, {2223, 1449}, {2254, 4801}, {2334, 673}, {2340, 391}, {3693, 4673}, {5089, 5342}, {6184, 4684}, {8694, 666}, {14439, 4742}

X(14627) =  X(3)X(6)∩X(49)X(51)

Barycentrics    a^2*(a^8 - 4*a^6*b^2 + 6*a^4*b^4 - 4*a^2*b^6 + b^8 - 4*a^6*c^2 + 5*a^4*b^2*c^2 + 3*a^2*b^4*c^2 - 4*b^6*c^2 + 6*a^4*c^4 + 3*a^2*b^2*c^4 + 6*b^4*c^4 - 4*a^2*c^6 - 4*b^2*c^6 + c^8) : :

In the plane of a triangle ABC, let
O = X(3) = circumcenter;
P = X(2070) = circumcircle-inverse of X(5);
(BCO) = circumcircle of B, C, O;
Pa = (BCO)-inverse of P, and define Pb and Pc cyclically.
The lines APa, BPb, CPc concur in X(14627). (Ivan Pavlov, May 22, 2022)

X(14627) lies on the cubic K439 and these lines: {3, 6}, {4, 13585}, {5, 195}, {24, 13321}, {25, 9704}, {30, 1199}, {49, 51}, {54, 143}, {110, 1173}, {140, 2889}, {155, 3851}, {156, 7545}, {265, 3574}, {323, 3628}, {381, 11441}, {382, 7592}, {394, 5070}, {399, 546}, {1154, 13434}, {1181, 3830}, {1353, 7403}, {1656, 1993}, {2904, 7507}, {2914, 11801}, {2937, 3060}, {2964, 11849}, {3090, 11004}, {3521, 12897}, {3526, 5422}, {3567, 11449}, {3843, 10982}, {5012, 10263}, {5076, 11456}, {5446, 5899}, {5562, 12316}, {5576, 13292}, {5898, 6153}, {6102, 11440}, {6288, 10112}, {6639, 11427}, {6640, 11433}, {7506, 9703}, {7512, 14449}, {7517, 11402}, {7722, 11559}, {9545, 12106}, {10110, 10540}, {11003, 13472}, {11597, 11800}, {12102, 12112}, {12370, 12902}

X(14627) = reflection of X(i) in X(j) for these {i,j}: {3, 13353}
X(14627) = circumcircle-inverse of X(11810)
X(14627) = X(1263)-Ceva conjugate of X(2070)
X(14627) = X(6)-Hirst inverse of X(11810)
X(14627) = X(512)-vertex conjugate of X(11810)
X(14627) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (5, 1994, 195), (49, 51, 13621), (52, 567, 3), (54, 143, 2070), (61, 62, 11063), (110, 1173, 10095), (156, 9781, 7545), (568, 578, 3), (569, 576, 6243), (569, 6243, 3), (1379, 1380, 11810), (1493, 10095, 110), (3311, 3312, 1609), (5012, 10263, 13564), (9781, 11422, 156), (13336, 13340, 3)

X(14628) =  X(2)X(2006)∩X(7)X(8046)

Barycentrics    b*(-a + b - c)*(a + b - c)*c*(-2*a + b + c)*(a^2 - a*b + b^2 - c^2)*(-a^2 + b^2 + a*c - c^2) : :

X(14628) lies on these lines: {2, 2006}, {7, 8046}, {55, 14204}, {57, 655}, {80, 497}, {226, 514}, {312, 1016}, {519, 14584}, {996, 1215}, {2222, 2726}

X(14628) = crossdifference of every pair of points on line {2361, 8648}
X(14628) = X(i)-isoconjugate of X(j) for these (i,j): {36, 2316}, {88, 2361}, {106, 2323}, {654, 901}, {1320, 7113}, {3257, 8648}, {4282, 4674}, {4511, 9456}
X(14628) = barycentric product X(i)*X(j) for these {i,j}: {75, 14584}, {655, 3762}, {1411, 3264}, {2006, 4358}
X(14628) = barycentric quotient X(i)/X(j) for these {i,j}: {44, 2323}, {80, 1320}, {214, 4996}, {519, 4511}, {655, 3257}, {900, 3738}, {902, 2361}, {1168, 1318}, {1317, 214}, {1319, 36}, {1404, 7113}, {1411, 106}, {1635, 654}, {1846, 1845}, {1877, 1870}, {1960, 8648}, {2006, 88}, {2161, 2316}, {2222, 901}, {3285, 4282}, {3762, 3904}, {3911, 3218}, {5298, 4973}, {12832, 11570}, {14584, 1}

X(14629) =  X(11)X(1168)∩X(36)X(80)

Barycentrics    (-a + b + c)*(a^2 - a*b + b^2 - c^2)*(a^2 - b^2 - a*c + c^2)*(a^5 - a^4*b - a^3*b^2 + 3*a^2*b^3 - 2*b^5 - a^4*c + 3*a^3*b*c - 3*a^2*b^2*c - 3*a*b^3*c + 4*b^4*c - a^3*c^2 - 3*a^2*b*c^2 + 6*a*b^2*c^2 - 2*b^3*c^2 + 3*a^2*c^3 - 3*a*b*c^3 - 2*b^2*c^3 + 4*b*c^4 - 2*c^5) : :
X(14629) = 2 X(80) + X(2222) = X(2716) - 4 X(12619)

X(14629) lies on these lines: {11, 1168}, {36, 80}, {2716, 12619}

X(14629) = barycentric quotient X(10703)/X(3218)

X(14630) =  X(2)X(6178)∩X(3)X(6)

Trilinears    e sin A - sin (A + ω) : :
Barycentrics    a^2*(b^2 + c^2 - Sqrt(a^4 - a^2*b^2 + b^4 - a^2*c^2 - b^2*c^2 + c^4)) : :

X(14630) lies on these lines: {2, 6178}, {3, 6}, {83, 3414}, {1078, 6189}, {1349, 7828}, {2040, 2543}, {5639, 5643}, {6190, 7760}, {7829, 14501}
p

X(14630) = isogonal conjugate of X(14632)
X(14630) = Brocard axis intercept, other than X(1379), of circle {X(1379),PU(1)}
X(14630) = inverse-in-circle {X(1687),X(1688),PU(1),PU(2)} of X(14631)
X(14630) = inverse-in-circle-O(61,62) of X(14631)
X(14630) = Brocard-circle-inverse of X(3558)
X(14630) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 6, 3558), (3,575,14631) (3, 3558, 1380), (6,39,14631), (6, 1340, 1380), (32, 182, 2558), (32, 2558, 1380), (32,5038,14631), (61,62,14631), (182,3398,14631), (371, 372, 1341), (1340, 3557, 13325), (1340, 3558, 3), (1342, 1343, 1380), (1687,1688,14631), (1689, 1690, 13326), (1691, 2029, 1379), (2559, 3557, 1379)

X(14631) =  X(2)X(6177)∩X(3)X(6)

Trilinears    e sin A + sin (A + ω) : :
Barycentrics    a^2*(b^2 + c^2 + Sqrt(a^4 - a^2*b^2 + b^4 - a^2*c^2 - b^2*c^2 + c^4)) : :

There are two anti-circummedial triangles, i.e. triangles of which ABC is the circummedial triangle. The 1st anti-circummedial triangle is the circumcevian triangle of the 1st focus, P(118), of the Steiner inellipse. The 2nd anti-circummedial triangle is the circumcevian triangle of the 2nd focus, U(118), of the Steiner inellipse. Let A'B'C' and A"B"C" be the 1st and 2nd anti-circummedial triangles, resp. Let A* be the crosssum of A' and A", and define B*and C* cyclically. The lines AA*, BB*, CC* concur in X(14631). Let A** be the crosspoint of A' and A", and define B** and C** cyclically. The lines AA**, BB**, CC** concur in X(14631). (Randy Hutson, November 2, 2017)

X(14631) lies on these lines: {2, 6177}, {3, 6}, {83, 3413}, {1078, 6190}, {1348, 7828}, {2039, 2542}, {5638, 5643}, {6189, 7760}, {7829, 14502}

X(14631) = isogonal conjugate of X(14633)
X(14631) = Brocard-circle-inverse of X(3557)
X(14631) = inverse-in-circle {X(1687),X(1688),PU(1),PU(2)} of X(14630)
X(14631) = inverse-in-circle-O(61,62) of X(14630)
X(14631) = Brocard axis intercept, other than X(1380), of circle {X(1380),PU(1)}
X(14631) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 6, 3557), (3,575,14630), (3, 3557, 1379), (6,39,14630), (6, 1341, 1379), (32, 182, 2559), (32, 2559, 1379), (32,5038,14630), (61,62,14630), (182,3398,14630), (371, 372, 1340), (1341, 3557, 3), (1341, 3558, 13326), (1342, 1343, 1379), (1689, 1690, 13325), (1687,1688,14630), (1691, 2028, 1380), (2558, 3558, 1380)

X(14632) =  MIDPOINT OF X(2028) AND X(2040)

Barycentrics    2*a^4 + b^4 + c^4 - (2*a^2 + b^2 + c^2)*Sqrt(a^4 - a^2*b^2 + b^4 - a^2*c^2 - b^2*c^2 + c^4) : :
X(14632) = X(4) + 3 X(1341) = 3 X(2) + X(3558) = 5 X(7786) - X(13325) = 3 X(262) + X(13326)

X(14632) lies on the Kiepert hyperbola, the cubic K054, and these lines: {2, 3558}, {4, 1341}, {5, 3413}, {6, 6178}, {39, 3414}, {83, 1380}, {140, 143}, {262, 13326}, {1506, 2028}, {1656, 6177}, {2558, 3407}, {6040, 14492}, {7786, 13325}, {9698, 14501}

X(14632) = midpoint of X(2028) and X(2040)
X(14632) = reflection of X(14633) in X(11272)
X(14632) = isogonal conjugate of X(14630)
X(14632) = X(i)-cross conjugate of X(j) for these (i,j): {2028, 3413}, {2040, 3414}

X(14633) =  MIDPOINT OF X(2029) AND X(2039)

Barycentrics    2*a^4 + b^4 + c^4 + (2*a^2 + b^2 + c^2)*Sqrt(a^4 - a^2*b^2 + b^4 - a^2*c^2 - b^2*c^2 + c^4) : :
X(14633) = X(4) + 3 X(1340) = 3 X(2) + X(3557) = 3 X(262) + X(13325) = 5 X(7786) - X(13326)

Let A'B'C' and A"B"C" be the 1st and 2nd anti-circummedial triangles, resp. Let A* be the cevapoint of A' and A", and define B* and C* cyclically. The lines AA*, BB*, CC* concur in X(14633). (Randy Hutson, November 2, 2017)

In the plane of a triangle ABC, let
F1 and F2 be the real foci of the Steiner inellipse
A1 = AF1∩BC
A2 = AF2∩BC
Ga = conic tangent to AF1 at A1, to AF2 at A2, and to F1F2
Ta = Ga∩F1F2
Ao = center of Ga, and define Bo and Co cyclically
The lines AAo, BBo, CCo concur in X(14633), and the lines ATa, BTb, CTc concur in X(6177). See X(6177) and X(14633). (Angel Montesdeoca, July 21, 2023)

X(14633) lies on the Kiepert hyperbola, the cubic K054, and these lines: {2, 3557}, {4, 1340}, {5, 3414}, {6, 6177}, {39, 3413}, {83, 1379}, {140, 143}, {262, 13325}, {1506, 2029}, {1656, 6178}, {2559, 3407}, {6039, 14492}, {7786, 13326}, {9698, 14502}

X(14633) = midpoint of X(2029) and X(2039)
X(14633) = reflection of X(14632) in X(11272)
X(14633) = isogonal conjugate of X(14631)
X(14633) = perspector of ABC and mid-triangle of 1st and 2nd anti-circummedial triangles
X(14633) = X(i)-cross conjugate of X(j) for these (i,j): {2029, 3414}, {2039, 3413}

X(14634) =  X(3)X(66)∩X(50)X(74)

Barycentrics    a^2*(2*a^10 - 4*a^8*b^2 + 2*a^6*b^4 - 2*a^4*b^6 + 4*a^2*b^8 - 2*b^10 - 4*a^8*c^2 + 12*a^6*b^2*c^2 - 3*a^4*b^4*c^2 - 2*a^2*b^6*c^2 - 3*b^8*c^2 + 2*a^6*c^4 - 3*a^4*b^2*c^4 - 4*a^2*b^4*c^4 + 5*b^6*c^4 - 2*a^4*c^6 - 2*a^2*b^2*c^6 + 5*b^4*c^6 + 4*a^2*c^8 - 3*b^2*c^8 - 2*c^10) : :

X(14634) lies on the cubic K488 and these lines: {3, 66}, {50, 74}, {378, 1990}, {1272, 2071}, {2781, 3284}, {5621, 11063}, {7464, 12253}

X(14635) =  REFLECTION OF X(10184) IN X(5)

Barycentrics    (a^2 b^2-b^4+a^2 c^2+2 b^2 c^2-c^4) (a^12-10 a^8 b^4+20 a^6 b^6-15 a^4 b^8+4 a^2 b^10-5 a^8 b^2 c^2+4 a^6 b^4 c^2+10 a^4 b^6 c^2-12 a^2 b^8 c^2+3 b^10 c^2-10 a^8 c^4+4 a^6 b^2 c^4+10 a^4 b^4 c^4+8 a^2 b^6 c^4-12 b^8 c^4+20 a^6 c^6+10 a^4 b^2 c^6+8 a^2 b^4 c^6+18 b^6 c^6-15 a^4 c^8-12 a^2 b^2 c^8-12 b^4 c^8+4 a^2 c^10+3 b^2 c^10) : :

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 26643.

X(14635) lies on these lines: {5,51}, {30,12012}, {1093,3851}, {3545,11197}, {8799,13322}

X(14635) = reflection of X(10184 in X(5)
X(14635) = X(10184)-of-Johnson-triangle

X(14636) =  15th HATZIPOLAKIS-MOSES-EULER POINT

Barycentrics    a (3 a^5 b+3 a^4 b^2-3 a^3 b^3-3 a^2 b^4+3 a^5 c+4 a^4 b c-5 a^2 b^3 c-3 a b^4 c+b^5 c+3 a^4 c^2-6 a^2 b^2 c^2-3 a b^3 c^2-3 a^3 c^3-5 a^2 b c^3-3 a b^2 c^3-2 b^3 c^3-3 a^2 c^4-3 a b c^4+b c^5) : :
X(14636) = 2 X(3) + X(9840) = X(500) - 4 X(13624)

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 26643.

X(14636) lies on these lines: {2,3}, {500,1193}, {511,3576}, {517,1962}, {524,11194}, {970,10470}, {1764,6176}, {2975,3578}, {3017,4276}, {3679,10434}, {4448,6002}

X(14636) = intersection of Euler lines of ABC and polar triangle of excircles radical circle

X(14637) =  X(6)X(1960)∩X(1646)X(8027)

Barycentrics    a^2*(2*a - b - c)^3*(b - c)^3 : :

X(14637) lies on these lines: {6, 1960}, {1646, 8027}

X(14637) = crossdifference of every pair of points on line {545, 1016}
X(14637) = isoconjugate of X(679) and X(6635)
X(14637) = barycentric product X(i)*X(j) for these {i,j}: {902, 14442}, {1017, 6550}, {2087, 3251}, {4370, 8661}
X(14637) = barycentric quotient X(1017)/X(6635)

X(14638) = X(253)X(523)∩X(3265)X(8057)

Barycentrics    (b - c)*(b + c)*(-a^2 + b^2 + c^2)^2*(a^4 - 2*a^2*b^2 + b^4 + 2*a^2*c^2 + 2*b^2*c^2 - 3*c^4)*(-a^4 - 2*a^2*b^2 + 3*b^4 + 2*a^2*c^2 - 2*b^2*c^2 - c^4) : :

X(14638) lies on these lines: {253, 523}, {3265, 8057}

X(14638) = X(525)-cross conjugate of X(3265)
X(14638) = X(i)-isoconjugate of X(j) for these (i,j): {112, 204}, {162, 3172}, {163, 6525}
X(14638) = barycentric product X(i)*X(j) for these {i,j}: {253, 3265}, {459, 4143}, {1073, 3267}
X(14638) = barycentric quotient X(i)/X(j) for these {i,j}: {253, 107}, {459, 6529}, {520, 154}, {523, 6525}, {525, 1249}, {647, 3172}, {656, 204}, {850, 14249}, {1073, 112}, {3265, 20}, {7068, 14308}, {8057, 3079}, {8611, 7156}, {14208, 1895}, {14379, 1576}

X(14639) =  X(4)X(32)∩X(5)X(99)

Barycentrics    a^8 - a^6*b^2 - a^4*b^4 + 3*a^2*b^6 - 2*b^8 - a^6*c^2 + 3*a^4*b^2*c^2 - 3*a^2*b^4*c^2 + 7*b^6*c^2 - a^4*c^4 - 3*a^2*b^2*c^4 - 10*b^4*c^4 + 3*a^2*c^6 + 7*b^2*c^6 - 2*c^8 : :
X(14639) = 2 X(4) + X(98) = 4 X(5) - X(99) = X(98) - 4 X(115) = X(4) + 2 X(115) = 2 X(114) + X(148) = 2 X(381) + X(671) = 4 X(620) - 7 X(3090) = 2 X(114) - 5 X(3091) = X(148) + 5 X(3091) = X(147) - 7 X(3832)

X(14639) is the center of inverse similitude of the 3rd and 4th Fermat-Dao equilateral triangles. (Randy Hutson, March 14, 2018)

X(14639) lies on these lines: {2, 9734}, {3, 10723}, {4, 32}, {5, 99}, {11, 13182}, {12, 13183}, {13, 5479}, {14, 5478}, {20, 6036}, {30, 9166}, {114, 148}, {119, 10769}, {147, 3832}, {262, 381}, {355, 7983}, {376, 5461}, {382, 12042}, {385, 13449}, {511, 14041}, {542, 3839}, {543, 3545}, {546, 6033}, {620, 3090}, {631, 6722}, {842, 14120}, {944, 11725}, {946, 7970}, {1352, 10754}, {1503, 6034}, {1916, 6248}, {1975, 8781}, {2023, 11257}, {2482, 5071}, {3023, 10896}, {3027, 10895}, {3044, 10539}, {3543, 6055}, {3583, 10053}, {3585, 10069}, {3843, 12188}, {3845, 11632}, {3851, 13188}, {4027, 10358}, {5056, 6721}, {5066, 8724}, {5182, 14561}, {5186, 7507}, {5198, 9861}, {5473, 6670}, {5474, 6669}, {5476, 8593}, {5480, 10753}, {5503, 7620}, {5691, 11710}, {5969, 10516}, {6249, 11606}, {6250, 14245}, {6251, 14231}, {6319, 10514}, {6320, 10515}, {6459, 8980}, {6460, 13967}, {6530, 10151}, {6568, 14240}, {6569, 14236}, {6811, 13773}, {6813, 13653}, {7395, 13175}, {7603, 7608}, {7687, 11005}, {7709, 11648}, {7741, 10089}, {7825, 12251}, {7902, 10359}, {7951, 10086}, {7989, 13174}, {8227, 11711}, {8754, 11596}, {8782, 10356}, {9478, 12122}, {9748, 11177}, {9864, 11599}

X(14639) = reflection of X(5182) in X(14561)
X(14639) = anticomplement of X(38748)
X(14639) = centroid of X(4)X(13)X(14)
X(14639) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4, 98, 10722), (4, 115, 98), (5, 6321, 99), (148, 3091, 114), (381, 671, 6054), (946, 13178, 7970), (3090, 13172, 620), (5480, 11646, 10753), (9862, 11623, 98), (10723, 14061, 3)

X(14640) =  X(3)X(107)∩X(5)X(53)

Barycentrics    (a^2*b^2 - b^4 + a^2*c^2 + 2*b^2*c^2 - c^4)*(a^12 - 4*a^10*b^2 + 6*a^8*b^4 - 4*a^6*b^6 + a^4*b^8 - 4*a^10*c^2 + 13*a^8*b^2*c^2 - 8*a^6*b^4*c^2 - 6*a^4*b^6*c^2 + 4*a^2*b^8*c^2 + b^10*c^2 + 6*a^8*c^4 - 8*a^6*b^2*c^4 + 10*a^4*b^4*c^4 - 4*a^2*b^6*c^4 - 4*b^8*c^4 - 4*a^6*c^6 - 6*a^4*b^2*c^6 - 4*a^2*b^4*c^6 + 6*b^6*c^6 + a^4*c^8 + 4*a^2*b^2*c^8 - 4*b^4*c^8 + b^2*c^10) : :
X(14640) = 2 X(6663) + X(11591)

X(14640) lies on these lines: {3, 107}, {5, 53}, {30, 5892}, {6663, 11591}

X(14641) =  X(3)X(1495)∩X(4)X(5892)

Barycentrics    a^2*(a^6*b^2 - 3*a^4*b^4 + 3*a^2*b^6 - b^8 + a^6*c^2 + 12*a^4*b^2*c^2 - 9*a^2*b^4*c^2 - 4*b^6*c^2 - 3*a^4*c^4 - 9*a^2*b^2*c^4 + 10*b^4*c^4 + 3*a^2*c^6 - 4*b^2*c^6 - c^8) : :
X(14641) = 6 X(143) - 7 X(389) = 3 X(51) - X(5073) = 8 X(143) - 7 X(5446) = 4 X(389) - 3 X(5446) = 3 X(376) - 2 X(5447) = 3 X(3534) - X(5562) = 11 X(3) - 9 X(5650) = 3 X(1216) - 2 X(5876) = 3 X(550) - X(5876)

X(14641) lies on these lines: {3, 1495}, {4, 5892}, {5, 10219}, {20, 6193}, {30, 143}, {51, 5073}, {52, 3529}, {140, 11017}, {185, 1657}, {376, 5447}, {382, 5462}, {511, 13491}, {548, 5907}, {550, 1216}, {1192, 7387}, {1204, 12083}, {1614, 10564}, {2071, 8718}, {3146, 9730}, {3522, 5891}, {3523, 11455}, {3524, 11439}, {3534, 5562}, {3567, 12002}, {3627, 9729}, {3845, 11695}, {3853, 5943}, {3858, 6688}, {4549, 12250}, {5059, 5890}, {5663, 12103}, {5889, 11001}, {6699, 13383}, {7512, 13445}, {7689, 11414}, {8703, 11793}, {9707, 11413}, {10263, 13382}, {11425, 12085}, {11576, 11802}, {11750, 12235}, {12102, 13363}, {12163, 12309}

X(14641) = midpoint of X(i) and X(j) for these {i,j}: {20, 10575}, {52, 3529}, {185, 1657}, {6241, 10625}, {12162, 12279}
X(14641) = reflection of X(i) in X(j) for these {i,j}: {382, 5462}, {1216, 550}, {3627, 9729}, {5876, 13348}, {5907, 548}, {10263, 13382}, {12162, 5447}, {13474, 140}, {13598, 13630}
X(14641) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (20, 6241, 10625), (376, 12162, 5447), (376, 12279, 12162), (550, 5876, 13348), (3522, 12290, 5891), (5876, 13348, 1216), (10575, 10625, 6241)

X(14642) =  X(6)X(64)∩X(41)X(1409)

Trilinears    (sin 2A)/(cos A - cos B cos C) : :
Barycentrics    a^4*(a^2 - b^2 - c^2)*(a^4 - 2*a^2*b^2 + b^4 + 2*a^2*c^2 + 2*b^2*c^2 - 3*c^4)*(a^4 + 2*a^2*b^2 - 3*b^4 - 2*a^2*c^2 + 2*b^2*c^2 + c^4) : :

X(14642) lies on the cubics K346, 378, and K429, and on these lines: {6, 64}, {41, 1409}, {184, 5065}, {193, 253}, {216, 11589}, {275, 459}, {393, 1562}, {394, 1073}, {417, 577}, {647, 2430}, {800, 1204}, {1249, 6225}, {1802, 3990}, {1988, 10311}, {3087, 6526}, {3284, 8798}, {5063, 14533}

X(14642) = isogonal conjugate of X(15466)
X(14642) = isotomic conjugate of the polar conjugate of X(33581)
X(14642) = X(i)-Ceva conjugate of X(j) for these (i,j): {1073, 14379}, {14379, 184}
X(14642) = X(32)-cross conjugate of X(184)
X(14642) = cevapoint of X(9306) and X(13346)
X(14642) = crosspoint of X(64) and X(1073)
X(14642) = crossdifference of every pair of points on line {1559, 8057}
X(14642) = crosssum of X(i) and X(j) for these (i,j): {2, 14361}, {20, 1249}
X(14642) = X(i)-isoconjugate of X(j) for these (i,j): {2, 1895}, {19, 14615}, {20, 92}, {63, 14249}, {75, 1249}, {76, 204}, {154, 1969}, {264, 610}, {286, 8804}, {304, 6525}, {331, 7070}, {459, 1097}, {561, 3172}, {811, 6587}, {823, 8057}, {1394, 7017}, {3213, 3596}, {6063, 7156}, {10152, 14206}
X(14642) = barycentric product X(i)*X(j) for these {i,j}: {3, 64}, {4, 14379}, {6, 1073}, {48, 2184}, {54, 8798}, {63, 2155}, {74, 11589}, {184, 253}, {212, 8809}, {459, 577}, {520, 1301}, {1092, 6526}, {13157, 14533}
X(14642) = barycentric quotient X(i)/X(j) for these {i,j}: {3, 14615}, {25, 14249}, {31, 1895}, {32, 1249}, {64, 264}, {184, 20}, {560, 204}, {1073, 76}, {1301, 6528}, {1501, 3172}, {1974, 6525}, {2155, 92}, {2184, 1969}, {2200, 8804}, {3049, 6587}, {8798, 311}, {9247, 610}, {9447, 7156}, {11589, 3260}, {14379, 69}, {14575, 154}

X(14643) = X(3)X(113)∩X(5)X(49)

Barycentrics    a^10 - 4*a^8*b^2 + 5*a^6*b^4 - a^4*b^6 - 2*a^2*b^8 + b^10 - 4*a^8*c^2 + 3*a^6*b^2*c^2 - 2*a^4*b^4*c^2 + 6*a^2*b^6*c^2 - 3*b^8*c^2 + 5*a^6*c^4 - 2*a^4*b^2*c^4 - 8*a^2*b^4*c^4 + 2*b^6*c^4 - a^4*c^6 + 6*a^2*b^2*c^6 + 2*b^4*c^6 - 2*a^2*c^8 - 3*b^2*c^8 + c^10 : :
X(14643) = 2 X(5) + X(110), X(3) + 2 X(113), X(74) - 4 X(140), 4 X(5) - X(265), 2 X(110) + X(265), 2 X(125) + X(399), X(146) + 5 X(631), X(4) + 2 X(1511), X(20) + 2 X(1539), 2 X(125) - 5 X(1656), X(399) + 5 X(1656)

Let Q be the quadrilateral ABCX(110). Taking the vertices 3 at a time yields four triangles whose nine-point centers are the vertices of a cyclic quadrilateral homothetic to Q at X(14643). (Randy Hutson, November 2, 2017)

Let P and U be the intersections of the 1st and 2nd Droz-Farny circles. Then X(14643) is the crosssum of P and U. (Randy Hutson, November 2, 2017)

X(14643) lies on the Walsmith rectangular hyperbola and these lines: {2, 5655}, {3, 113}, {4, 1511}, {5, 49}, {11, 10088}, {12, 10091}, {20, 1539}, {35, 12374}, {36, 12373}, {74, 140}, {119, 12905}, {125, 399}, {141, 9970}, {146, 631}, {156, 12419}, {355, 11720}, {381, 5642}, {468, 3581}, {498, 3024}, {499, 3028}, {541, 5054}, {542, 5050}, {546, 10733}, {547, 9140}, {549, 10706}, {550, 10721}, {568, 5654}, {946, 12778}, {1112, 3542}, {1216, 13417}, {1351, 5181}, {1352, 6593}, {1385, 12368}, {1482, 11723}, {1495, 7574}, {1503, 2072}, {1568, 2070}, {1657, 13202}, {1986, 7505}, {2771, 10202}, {2854, 14561}, {2888, 11702}, {2914, 3519}, {2931, 7506}, {2948, 8227}, {3070, 10820}, {3071, 10819}, {3090, 3448}, {3091, 10113}, {3311, 8998}, {3312, 13990}, {3523, 12244}, {3526, 6699}, {3541, 12133}, {3547, 13416}, {3548, 5656}, {3549, 12358}, {3567, 12273}, {3589, 11579}, {3628, 10264}, {3818, 7579}, {3843, 12295}, {3845, 11694}, {3851, 7687}, {5067, 12317}, {5070, 6053}, {5071, 9143}, {5095, 11898}, {5432, 10065}, {5433, 10081}, {5448, 12893}, {5465, 8724}, {5476, 5648}, {5504, 9820}, {5562, 11557}, {5690, 7978}, {5876, 7722}, {5878, 11598}, {5891, 10628}, {5901, 7984}, {5907, 11562}, {6639, 7723}, {6642, 12168}, {7393, 13171}, {7395, 12412}, {7507, 12140}, {7528, 12319}, {7529, 12310}, {7691, 11805}, {7699, 10546}, {7731, 11444}, {7741, 12904}, {7951, 12903}, {7989, 12407}, {7999, 13201}, {9033, 11911}, {9129, 10748}, {9306, 10254}, {9956, 11699}, {10104, 13193}, {10201, 12824}, {10255, 10539}, {10356, 12501}, {10358, 12201}, {10514, 12803}, {10515, 12804}, {10576, 12376}, {10577, 12375}, {10625, 11807}, {10896, 12896}, {11064, 11799}, {11591, 12219}, {11693, 14269}, {12901, 14130}, {13198, 13353}

X(14643) = anticomplement of X(34128)
X(14643) = orthocenter of X(110)X(113)X(125)
X(14643) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 113, 7728), (4, 1511, 12121), (5, 110, 265), (5, 10272, 110), (110, 12228, 49), (113, 5972, 3), (125, 12900, 1656), (146, 631, 12041), (355, 11720, 12898), (399, 1656, 125), (2948, 8227, 12261), (3091, 12383, 10113), (3526, 10620, 6699), (3851, 12902, 7687), (5876, 11561, 7722), (9956, 11699, 13211)
X(14643) = barycentric product X(323)*X(10688)
X(14643) = barycentric quotient X(10688)/X(94)

X(14644) = X(4)X(74)∩X(5)X(49)

Barycentrics    -a^10 + a^8*b^2 + a^6*b^4 + a^4*b^6 - 4*a^2*b^8 + 2*b^10 + a^8*c^2 - 3*a^6*b^2*c^2 - a^4*b^4*c^2 + 9*a^2*b^6*c^2 - 6*b^8*c^2 + a^6*c^4 - a^4*b^2*c^4 - 10*a^2*b^4*c^4 + 4*b^6*c^4 + a^4*c^6 + 9*a^2*b^2*c^6 + 4*b^4*c^6 - 4*a^2*c^8 - 6*b^2*c^8 + 2*c^10 : :
X(14644) = 2 X(4) + X(74) = 4 X(5) - X(110) = X(74) - 4 X(125) = X(4) + 2 X(125) = 2 X(5) + X(265) = X(110) + 2 X(265) = X(895) + 2 X(1352) = 2 X(1511) - 5 X(1656) = 2 X(113) - 5 X(3091) = X(477) - 4 X(3154) = 2 X(113) + X(3448)

Let A'B'C' be the orthocentroidal triangle. X(14644) is the radical center of the circumcircles of AB'C', BC'A', and CA'B'. (Randy Hutson, November 2, 2017)

X(14644) lies on these lines: {3, 10113}, {4, 74}, {5, 49}, {6, 7699}, {11, 12903}, {12, 12904}, {20, 6699}, {51, 10628}, {52, 12219}, {67, 5480}, {113, 3091}, {115, 6794}, {119, 10778}, {140, 12121}, {146, 3832}, {186, 13851}, {355, 7984}, {381, 5640}, {382, 12041}, {389, 7722}, {399, 3851}, {403, 1503}, {477, 3154}, {498, 12896}, {523, 5627}, {541, 3839}, {542, 3545}, {546, 7728}, {578, 3043}, {631, 6723}, {895, 1352}, {944, 11735}, {946, 7978}, {974, 6241}, {1112, 7507}, {1173, 3574}, {1199, 12227}, {1350, 6698}, {1511, 1656}, {1539, 3843}, {1550, 14120}, {1568, 5965}, {1598, 13171}, {1614, 13198}, {1853, 11455}, {1986, 3567}, {2771, 5927}, {2854, 10516}, {2914, 12234}, {2931, 7503}, {2948, 7989}, {3024, 10896}, {3028, 10895}, {3047, 10539}, {3090, 5972}, {3311, 13915}, {3312, 13979}, {3518, 13289}, {3542, 12140}, {3580, 10297}, {3583, 10065}, {3585, 10081}, {3818, 11579}, {3855, 12317}, {5012, 10254}, {5056, 12900}, {5066, 5655}, {5071, 5642}, {5072, 5609}, {5198, 9919}, {5462, 11562}, {5504, 9927}, {5562, 11800}, {5656, 11457}, {5691, 11709}, {5876, 13358}, {5889, 7723}, {5901, 12898}, {6143, 13403}, {6247, 11744}, {6459, 8994}, {6460, 13969}, {6811, 13774}, {6813, 13654}, {6816, 12319}, {7395, 12310}, {7505, 12289}, {7529, 12412}, {7686, 10693}, {7732, 10514}, {7733, 10515}, {7741, 10091}, {7951, 10088}, {8227, 11720}, {9753, 9769}, {9880, 11006}, {9956, 12778}, {10104, 12201}, {10110, 13417}, {10117, 10594}, {10293, 13603}, {10356, 13210}, {10358, 13193}, {10576, 10819}, {10577, 10820}, {11412, 12358}, {11432, 12165}, {11479, 12168}, {11806, 12162}, {12133, 12290}, {12284, 12825}, {12368, 13605}, {12812, 13392}, {12893, 14118}

X(14644) = anticomplement of X(38793)
X(14644) = orthocentroidal-circle inverse of X(5890)
X(14644) = crossdifference of every pair of points on line {1636, 2081}
X(14644) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 10113, 10733), (4, 74, 10721), (4, 125, 74), (4, 12244, 13202), (5, 265, 110), (5, 11801, 265), (67, 5480, 10752), (110, 13434, 12228), (113, 3448, 14094), (125, 7687, 4), (355, 12261, 7984), (381, 9140, 10706), (546, 10264, 7728), (946, 13211, 7978), (974, 12292, 6241), (1656, 12902, 1511), (1986, 11746, 3567), (3090, 12383, 5972), (3091, 3448, 113), (3567, 12281, 1986), (3843, 10620, 1539), (6699, 12295, 20), (7723, 12236, 5889), (7731, 9781, 1112), (8227, 12407, 11720)

X(14645) = X(2)X(2987)∩X(6)X(620)

Barycentrics    2*a^6 - 4*a^4*b^2 + 5*a^2*b^4 - b^6 - 4*a^4*c^2 - b^4*c^2 + 5*a^2*c^4 - b^2*c^4 - c^6 : :

X(14645) lies on the cubic K250 and these lines: {2, 2987}, {6, 620}, {30, 511}, {32, 1992}, {69, 115}, {99, 193}, {114, 1351}, {126, 6792}, {141, 6722}, {315, 671}, {325, 5107}, {487, 9894}, {488, 9892}, {576, 7764}, {591, 13873}, {597, 6680}, {599, 626}, {1270, 13653}, {1271, 13773}, {1570, 6393}, {1991, 13926}, {2031, 6390}, {3620, 14061}, {3629, 5026}, {5052, 5976}, {5108, 6719}, {5162, 9891}, {5468, 10418}, {5503, 7612}, {5921, 10723}, {6036, 13468}, {6055, 8667}, {6309, 7890}, {6321, 11898}, {7751, 11623}, {7758, 9890}, {7759, 11477}, {7810, 13085}, {7834, 10542}, {7837, 9764}, {7838, 8149}, {9169, 12036}, {9742, 9877}, {9753, 9770}, {10350, 12191}, {10991, 14023}, {13087, 13088}, {13875, 13929}, {13876, 13928}

X(14645) = isogonal conjugate of X(14659)
X(14645) = crossdifference of every pair of points on line {6, 2872}
X(14645) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (69, 10754, 115), (99, 193, 5477), (5108, 6791, 6719), (6792, 9146, 126)

X(14646) =  X(4)X(46)∩X(84)X(6361)

Barycentrics    5*a^6 - 6*a^5*b - 7*a^4*b^2 + 8*a^3*b^3 + 3*a^2*b^4 - 2*a*b^5 - b^6 - 6*a^5*c + 22*a^4*b*c - 8*a^3*b^2*c - 8*a^2*b^3*c - 2*a*b^4*c + 2*b^5*c - 7*a^4*c^2 - 8*a^3*b*c^2 + 10*a^2*b^2*c^2 + 4*a*b^3*c^2 + b^4*c^2 + 8*a^3*c^3 - 8*a^2*b*c^3 + 4*a*b^2*c^3 - 4*b^3*c^3 + 3*a^2*c^4 - 2*a*b*c^4 + b^2*c^4 - 2*a*c^5 + 2*b*c^5 - c^6 : :
X(14646) = X(4) - 4 X(1158) = 4 X(3579) - X(6223) = 7 X(3528) - 4 X(6261) = 2 X(84) + X(6361) = 2 X(144) + X(10307) = 8 X(5450) - 5 X(10595) = 2 X(5493) + X(10864) = 2 X(40) + X(12246) = 11 X(3525) - 8 X(12608) = 5 X(4) - 8 X(12616) = 5 X(1158) - 2 X(12616)

X(14646) lies one these lines: {4, 46}, {40, 12246}, {84, 6361}, {144, 6244}, {165, 5658}, {376, 5918}, {497, 1768}, {515, 4677}, {516, 3928}, {553, 5603}, {631, 3683}, {971, 3681}, {1056, 2096}, {2800, 7967}, {2950, 12115}, {3525, 12608}, {3528, 6261}, {3579, 6223}, {3651, 12330}, {3929, 5657}, {4421, 5851}, {5274, 13226}, {5435, 8166}, {5450, 10595}, {5493, 10864}, {5759, 10860}

X(14646) = reflection of X(5658) in X(165)
X(14646) = {X(1709,X(3474)}-harmonic conjugate of X(4)

X(14647) =  X(1)X(6705)∩X(4)X(46)

Barycentrics    a^7 + a^6*b - 5*a^5*b^2 - a^4*b^3 + 7*a^3*b^4 - a^2*b^5 - 3*a*b^6 + b^7 + a^6*c + 2*a^5*b*c + 5*a^4*b^2*c - 4*a^3*b^3*c - 5*a^2*b^4*c + 2*a*b^5*c - b^6*c - 5*a^5*c^2 + 5*a^4*b*c^2 - 6*a^3*b^2*c^2 + 6*a^2*b^3*c^2 + 3*a*b^4*c^2 - 3*b^5*c^2 - a^4*c^3 - 4*a^3*b*c^3 + 6*a^2*b^2*c^3 - 4*a*b^3*c^3 + 3*b^4*c^3 + 7*a^3*c^4 - 5*a^2*b*c^4 + 3*a*b^2*c^4 + 3*b^3*c^4 - a^2*c^5 + 2*a*b*c^5 - 3*b^2*c^5 - 3*a*c^6 - b*c^6 + c^7 : :

X(14647) = 2 X(10) + X(84) = X(4) + 2 X(1158) = X(2550) + 2 X(3358) = X(944) - 4 X(5450) = 2 X(3579) + X(5787) = X(40) + 2 X(6245) = 5 X(5818) - 2 X(6256) = 5 X(1698) - 2 X(6260) = 5 X(631) - 2 X(6261) = X(1490) - 4 X(6684) = X(1) - 4 X(6705) = 4 X(1125) - X(7971) = 5 X(1698) + X(7992) = 2 X(6260) + X(7992) = X(6223) - 7 X(9780) = 2 X(6684) + X(9948) = X(1490) + 2 X(9948) = X(6259) - 4 X(9956) = X(2950) + 2 X(10265) = X(3189) - 4 X(11248) = X(9799) + 2 X(11500) = X(8) + 2 X(12114) = 5 X(5818) + X(12246) = 2 X(6256) + X(12246) = X(9803) + 2 X(12332) = 7 X(3090) - 4 X(12608) = X(4) - 4 X(12616) = X(1158) + 2 X(12616) = 2 X(11362) + X(12650) = 2 X(9943) + X(12664) = 4 X(5777) - X(12666) = 4 X(10) - X(12667) = 2 X(84) + X(12667)

X(14647) lies on the cubic K815 and these lines: {1, 6705}, {2, 6001}, {3, 1610}, {4, 46}, {7, 7680}, {8, 6909}, {10, 84}, {20, 5086}, {35, 944}, {40, 4847}, {55, 5768}, {65, 6847}, {104, 3476}, {165, 376}, {278, 1735}, {281, 1765}, {355, 6948}, {405, 12330}, {517, 5770}, {601, 5716}, {631, 6261}, {938, 11496}, {946, 3339}, {960, 6926}, {999, 13226}, {1071, 3085}, {1125, 7971}, {1210, 12705}, {1329, 5811}, {1478, 1768}, {1490, 6684}, {1519, 10589}, {1698, 6260}, {1836, 6844}, {2550, 3358}, {2551, 7330}, {2800, 5603}, {2950, 10265}, {3086, 12672}, {3090, 12608}, {3189, 11248}, {3427, 3428}, {3485, 6833}, {3486, 6906}, {3487, 5884}, {3579, 5787}, {3812, 6846}, {3820, 5779}, {3869, 6890}, {4295, 6831}, {4312, 5924}, {4511, 6966}, {4640, 6987}, {4915, 11362}, {5175, 11826}, {5552, 12528}, {5698, 6827}, {5704, 7681}, {5777, 12666}, {5817, 10175}, {5818, 6256}, {5842, 9778}, {5880, 6843}, {5887, 6891}, {6223, 9780}, {6259, 9956}, {6838, 9961}, {6848, 12688}, {6855, 12609}, {6864, 12617}, {6865, 12514}, {6908, 9943}, {6939, 12686}, {6943, 11415}, {6956, 12047}, {6988, 12520}, {7411, 9799}, {7682, 11372}, {9803, 12332}, {10039, 10085}

X(14647) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (10, 84, 12667), (1158, 12616, 4), (1478, 1768, 2096), (1698, 7992, 6260), (1709, 1737, 4), (5818, 12246, 6256), (6684, 9948, 1490)
X(14647) = X(521)-gimel conjugate of X(2096)

X(14648) =  X(8)X(20)∩X(109)X(3241)

Barycentrics    3*a^7 + a^6*b - 11*a^5*b^2 - a^4*b^3 + 13*a^3*b^4 - a^2*b^5 - 5*a*b^6 + b^7 + a^6*c + 8*a^5*b*c + 7*a^4*b^2*c - 8*a^3*b^3*c - 9*a^2*b^4*c + b^6*c - 11*a^5*c^2 + 7*a^4*b*c^2 - 10*a^3*b^2*c^2 + 10*a^2*b^3*c^2 + 5*a*b^4*c^2 - b^5*c^2 - a^4*c^3 - 8*a^3*b*c^3 + 10*a^2*b^2*c^3 - b^4*c^3 + 13*a^3*c^4 - 9*a^2*b*c^4 + 5*a*b^2*c^4 - b^3*c^4 - a^2*c^5 - b^2*c^5 - 5*a*c^6 + b*c^6 + c^7 : :

X(14648) lies on the cubic K815 and these lines: {8, 20}, {109, 3241}

leftri

Singular Foci of Cubics: X(14649)-X(14706)

rightri

Let O(P) denote the orthopivotal cubic of a point P, as defined at Orthopivotal cubics. Five points on this cubic are A, B, C, X(13), and X(14). The singular focus of O(P), denoted by Psi(P) defines an involutory mapping P → Psi(P), and Psi(P) is called the Psi-transform of P. The centers X(14649)-X(14706) involve these orthopivotal cubics for these points: X(14651), X(14656), X(14660), X(14704), X(14705), X(14706). The others involve circular cubics. See Singular Focus of Circular Cubics in Bernard Gibert's CTC.

X(14649) =  SINGULAR FOCUS OF THE CUBIC K022

Barycentrics    a^2*(3*a^12 - 7*a^10*b^2 + 4*a^8*b^4 + 2*a^6*b^6 - 5*a^4*b^8 + 5*a^2*b^10 - 2*b^12 - 7*a^10*c^2 + 7*a^8*b^2*c^2 - 2*a^6*b^4*c^2 + 4*a^4*b^6*c^2 - 3*a^2*b^8*c^2 + b^10*c^2 + 4*a^8*c^4 - 2*a^6*b^2*c^4 - 2*a^4*b^4*c^4 - 2*a^2*b^6*c^4 + 2*b^8*c^4 + 2*a^6*c^6 + 4*a^4*b^2*c^6 - 2*a^2*b^4*c^6 - 2*b^6*c^6 - 5*a^4*c^8 - 3*a^2*b^2*c^8 + 2*b^4*c^8 + 5*a^2*c^10 + b^2*c^10 - 2*c^12) : :

X(14649) lies on the Yiu circle and these lines: {2, 2794}, {3, 1177}, {112, 186}, {574, 10766}, {1297, 8722}, {3455, 5622}, {7577, 10735}, {9126, 9517}, {11643, 14246}

X(14649) = circumcircle-inverse of X(9970)
X(14649) = Psi-transform of X(6800)
X(14649) = X(9517)-vertex conjugate of X(9970)

X(14650) =  SINGULAR FOCUS OF THE CUBIC K043

Barycentrics    a^2*(2*a^8 - 7*a^6*b^2 + a^4*b^4 + 7*a^2*b^6 - 3*b^8 - 7*a^6*c^2 + 28*a^4*b^2*c^2 - 20*a^2*b^4*c^2 + 11*b^6*c^2 + a^4*c^4 - 20*a^2*b^2*c^4 - 8*b^4*c^4 + 7*a^2*c^6 + 11*b^2*c^6 - 3*c^8) : :
X(14650) = 3 X(3) - X(1296) = 3 X(111) + X(1296) = X(5512) - 3 X(9172) = 3 X(10246) - X(10704) = 3 X(5054) - X(10717) = 3 X(381) - X(10734) = 3 X(5050) - X(10765) = 3 X(111) - X(11258) = 3 X(3) + X(11258) = 5 X(631) - X(14360)

X(14650) lies on these lines: {2, 10748}, {3, 111}, {5, 6719}, {30, 5512}, {35, 6019}, {36, 3325}, {126, 140}, {182, 1511}, {381, 10734}, {517, 11721}, {543, 549}, {631, 14360}, {2780, 9208}, {2793, 12042}, {5050, 10765}, {5054, 10717}, {5663, 9129}, {6088, 9126}, {6409, 11835}, {6410, 11836}, {8585, 10204}, {10246, 10704}

X(14650) = complement X(10748)
X(14650) = midpoint of X(i) and X(j) for these {i,j}: {3, 111}, {1296, 11258}
X(14650) = reflection of X(i) in X(j) for these {i,j}: {5, 6719}, {126, 140}
X(14650) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 11258, 1296), (111, 1296, 11258)
X(14650) = circumcircle-inverse of X(11258)
X(14650) = X(6088)-vertex conjugate of X(11258)
X(14650) = QA-P9 (QA-Miquel Center) of quadrangle ABCX(2)
X(14650) = X(5)-of-4th-anti-Brocard triangle

X(14651) =  SINGULAR FOCUS OF THE CUBIC K062

Barycentrics    a^8 - a^6*b^2 + 2*a^4*b^4 - 3*a^2*b^6 + b^8 - a^6*c^2 - 3*a^4*b^2*c^2 + 3*a^2*b^4*c^2 - 5*b^6*c^2 + 2*a^4*c^4 + 3*a^2*b^2*c^4 + 8*b^4*c^4 - 3*a^2*c^6 - 5*b^2*c^6 + c^8 : :
X(14651) = X(4) + 2 X(98) = X(4) - 4 X(115) = X(98) + 2 X(115) = 4 X(5) - X(147) = 2 X(3) + X(148) = 2 X(99) - 5 X(631) = X(376) + 2 X(671) = 4 X(114) - 7 X(3090) = 8 X(620) - 11 X(3525) = 5 X(5071) - 8 X(5461) = 5 X(3091) + X(5984) = 5 X(3091) - 2 X(6033) = X(5984) + 2 X(6033) = 5 X(631) - 8 X(6036) = X(99) - 4 X(6036) = 5 X(5071) - 2 X(6054) = 4 X(5461) - X(6054) = X(376) - 4 X(6055) = X(671) + 2 X(6055) = X(2698) + 2 X(6071) = X(20) + 2 X(6321) = 13 X(5067) - 16 X(6722) = 2 X(14) + X(6770) = X(617) - 4 X(6771) = 2 X(13) + X(6773) = X(616) - 4 X(6774) = 4 X(549) - X(8591) = 5 X(2) - 2 X(8724) = 2 X(946) + X(9860) = 4 X(98) - X(9862) = 2 X(4) + X(9862) = 8 X(115) + X(9862) = 5 X(5818) - 2 X(9864) = 2 X(7970) - 5 X(10595) = 5 X(4) - 2 X(10722) = 10 X(115) - X(10722) = 5 X(98) + X(10722) = 5 X(9862) + 4 X(10722) = X(3529) + 2 X(10723) = 5 X(9862) - 8 X(10991) = 5 X(98) - 2 X(10991) = 5 X(115) + X(10991) = X(10722) + 2 X(10991) = 5 X(4) + 4 X(10991) = 13 X(10299) - 4 X(10992) = 2 X(381) + X(11177) = X(40) + 2 X(11599), X(9862) - 16 X(11623), X(10991) - 10 X(11623), X(98) - 4 X(11623), X(115) + 2 X(11623), X(4) + 8 X(11623), X(2) + 2 X(11632)

X(14651) lies on these lines: {2, 2782}, {3, 148}, {4, 32}, {5, 147}, {13, 6773}, {14, 6770}, {20, 6321}, {30, 8859}, {40, 11599}, {99, 631}, {114, 3090}, {140, 13188}, {230, 11676}, {262, 5309}, {376, 671}, {381, 9755}, {388, 10069}, {403, 6761}, {497, 10053}, {511, 14568}, {542, 3545}, {543, 3524}, {549, 8591}, {616, 6774}, {617, 6771}, {620, 3525}, {944, 11710}, {946, 9860}, {1916, 12251}, {2023, 5286}, {2478, 5985}, {2698, 6071}, {2784, 5587}, {3023, 3086}, {3027, 3085}, {3088, 5186}, {3089, 12131}, {3091, 5984}, {3406, 11606}, {3529, 10723}, {3618, 12177}, {4027, 10359}, {4293, 13182}, {4294, 13183}, {5067, 6722}, {5071, 5461}, {5152, 11185}, {5218, 10086}, {5818, 9864}, {5965, 7809}, {5969, 10519}, {5986, 6997}, {6248, 7828}, {6319, 10517}, {6320, 10518}, {6655, 10104}, {6684, 13174}, {6776, 11646}, {7288, 10089}, {7746, 11257}, {7836, 13108}, {7851, 9478}, {7856, 10358}, {7970, 10595}, {7983, 12245}, {8703, 12355}, {8782, 10357}, {9140, 11656}, {9861, 10594}, {10070, 10385}, {10299, 10992}, {10323, 13175}, {10590, 12184}, {10591, 12185}, {10596, 12189}, {10597, 12190}, {10598, 12182}, {10599, 12183}, {10769, 13199}, {14482, 14494}

X(14651) = reflection of X(3545) in X(9166)
X(14651) = Psi-transform of X(51)
X(14651) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 11632, 12243), (3, 148, 13172), (4, 98, 9862), (4, 7735, 10788), (5, 12188, 147), (98, 115, 4), (98, 10722, 10991), (99, 6036, 631), (114, 14061, 3090), (115, 11623, 98), (671, 6055, 376), (3091, 5984, 6033), (5461, 6054, 5071), (6321, 12042, 20), (7970, 11725, 10595), (11710, 13178, 944)

X(14652) =  SINGULAR FOCUS OF THE CUBIC K073

Barycentrics    a^2*(a^10 - 3*a^8*b^2 + 4*a^6*b^4 - 4*a^4*b^6 + 3*a^2*b^8 - b^10 - 3*a^8*c^2 + 4*a^6*b^2*c^2 - a^4*b^4*c^2 - 2*a^2*b^6*c^2 + 2*b^8*c^2 + 4*a^6*c^4 - a^4*b^2*c^4 + a^2*b^4*c^4 - b^6*c^4 - 4*a^4*c^6 - 2*a^2*b^2*c^6 - b^4*c^6 + 3*a^2*c^8 + 2*b^2*c^8 - c^10) : :

X(14652) lies on lines these lines: {3, 2888}, {22, 11671}, {23, 137}, {94, 96}, {98, 930}, {184, 13504}, {186, 2970}, {1263, 2937}, {2070, 12026}, {7525, 13512}

X(14652) = circumcircle-inverse of X(3448)

X(14653) =  SINGULAR FOCUS OF THE CUBIC K088

Barycentrics    7*a^10 - 20*a^8*b^2 + a^6*b^4 + 19*a^4*b^6 - 8*a^2*b^8 + b^10 - 20*a^8*c^2 + 49*a^6*b^2*c^2 - 24*a^4*b^4*c^2 - 2*a^2*b^6*c^2 + b^8*c^2 + a^6*c^4 - 24*a^4*b^2*c^4 + 12*a^2*b^4*c^4 - 2*b^6*c^4 + 19*a^4*c^6 - 2*a^2*b^2*c^6 - 2*b^4*c^6 - 8*a^2*c^8 + b^2*c^8 + c^10 : :

X(14653) lies on these lines: {2, 10748}, {3, 67}, {30, 9759}, {110, 6093}, {381, 10418}, {2854, 7618}, {6593, 13608}

X(14653) = circumcircle-inverse of X(9966)
X(14653) = X(690)-vertex conjugate of X(9966)

X(14654) = SINGULAR FOCUS OF THE CUBIC K103

Barycentrics    3*a^10 - 9*a^8*b^2 - 2*a^6*b^4 + 10*a^4*b^6 - a^2*b^8 - b^10 - 9*a^8*c^2 + 31*a^6*b^2*c^2 - 16*a^4*b^4*c^2 - 3*a^2*b^6*c^2 + 5*b^8*c^2 - 2*a^6*c^4 - 16*a^4*b^2*c^4 + 8*a^2*b^4*c^4 - 4*b^6*c^4 + 10*a^4*c^6 - 3*a^2*b^2*c^6 - 4*b^4*c^6 - a^2*c^8 + 5*b^2*c^8 - c^10 : :
X(14654) = 4 X(126) - 5 X(631) = 3 X(376) - 2 X(1296) = 3 X(4) - 4 X(5512) = 3 X(111) - 2 X(5512) = 7 X(3090) - 8 X(6719) = 3 X(3545) - 4 X(9172) = 3 X(7967) - 2 X(10704) = 3 X(3524) - 2 X(10717) = 3 X(4) - 2 X(10734) = 3 X(111) - X(10734) = 3 X(5603) - 4 X(11721)

X(14654) lies on these lines: {2, 10748}, {3, 14360}, {4, 111}, {30, 11258}, {98, 376}, {126, 631}, {974, 2854}, {2780, 12244}, {2793, 9862}, {2805, 13199}, {2830, 12248}, {2847, 5667}, {3090, 6719}, {3325, 4293}, {3524, 10717}, {3545, 9172}, {4294, 6019}, {5603, 11721}, {7967, 10704}, {11568, 11636}

X(14654) = reflection of X(i) in X(j) for these {i,j}: {4, 111}, {10734, 5512}, {14360, 3}
X(14654) = anticomplement X(10748)
X(14654) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (111, 10734, 5512), (5512, 10734, 4)

X(14655) =  SINGULAR FOCUS OF THE CUBIC K108

Barycentrics    a^2*(a^14 - 4*a^12*b^2 + 2*a^10*b^4 + 7*a^8*b^6 - 7*a^6*b^8 - 2*a^4*b^10 + 4*a^2*b^12 - b^14 - 4*a^12*c^2 + 15*a^10*b^2*c^2 - 12*a^8*b^4*c^2 - 3*a^6*b^6*c^2 + 13*a^4*b^8*c^2 - 12*a^2*b^10*c^2 + 3*b^12*c^2 + 2*a^10*c^4 - 12*a^8*b^2*c^4 + 10*a^6*b^4*c^4 - 7*a^4*b^6*c^4 + 8*a^2*b^8*c^4 - b^10*c^4 + 7*a^8*c^6 - 3*a^6*b^2*c^6 - 7*a^4*b^4*c^6 - b^8*c^6 - 7*a^6*c^8 + 13*a^4*b^2*c^8 + 8*a^2*b^4*c^8 - b^6*c^8 - 2*a^4*c^10 - 12*a^2*b^2*c^10 - b^4*c^10 + 4*a^2*c^12 + 3*b^2*c^12 - c^14) : :

X(14655) lies on these lines: {3, 126}, {24, 1560}, {186, 2770}, {2373, 7488}, {12584, 13289}

X(14655) = circumcircle-inverse of X(10748)

X(14656) =  SINGULAR FOCUS OF THE CUBIC K112

Barycentrics    a^2*(a^16 - 5*a^14*b^2 + 10*a^12*b^4 - 9*a^10*b^6 + 9*a^6*b^10 - 10*a^4*b^12 + 5*a^2*b^14 - b^16 - 5*a^14*c^2 + 18*a^12*b^2*c^2 - 25*a^10*b^4*c^2 + 18*a^8*b^6*c^2 - 14*a^6*b^8*c^2 + 19*a^4*b^10*c^2 - 16*a^2*b^12*c^2 + 5*b^14*c^2 + 10*a^12*c^4 - 25*a^10*b^2*c^4 + 21*a^8*b^4*c^4 - 4*a^6*b^6*c^4 - 11*a^4*b^8*c^4 + 21*a^2*b^10*c^4 - 12*b^12*c^4 - 9*a^10*c^6 + 18*a^8*b^2*c^6 - 4*a^6*b^4*c^6 + 4*a^4*b^6*c^6 - 10*a^2*b^8*c^6 + 19*b^10*c^6 - 14*a^6*b^2*c^8 - 11*a^4*b^4*c^8 - 10*a^2*b^6*c^8 - 22*b^8*c^8 + 9*a^6*c^10 + 19*a^4*b^2*c^10 + 21*a^2*b^4*c^10 + 19*b^6*c^10 - 10*a^4*c^12 - 16*a^2*b^2*c^12 - 12*b^4*c^12 + 5*a^2*c^14 + 5*b^2*c^14 - c^16) : :

X(14656) lies on the tangential circle, the curve Q041, and these lines: {3, 137}, {22, 5966}, {24, 933}, {195, 568}, {2937, 12011}

X(14656) = circumcircle-inverse of X(137)
X(14656) = Psi-transform of X(54)

X(14657) =  SINGULAR FOCUS OF THE CUBIC K113

Barycentrics    a^2*(a^14 - 5*a^12*b^2 + 3*a^10*b^4 + 9*a^8*b^6 - 9*a^6*b^8 - 3*a^4*b^10 + 5*a^2*b^12 - b^14 - 5*a^12*c^2 + 27*a^10*b^2*c^2 - 29*a^8*b^4*c^2 - 2*a^6*b^6*c^2 + 29*a^4*b^8*c^2 - 25*a^2*b^10*c^2 + 5*b^12*c^2 + 3*a^10*c^4 - 29*a^8*b^2*c^4 + 38*a^6*b^4*c^4 - 26*a^4*b^6*c^4 + 19*a^2*b^8*c^4 - 5*b^10*c^4 + 9*a^8*c^6 - 2*a^6*b^2*c^6 - 26*a^4*b^4*c^6 + 2*a^2*b^6*c^6 + b^8*c^6 - 9*a^6*c^8 + 29*a^4*b^2*c^8 + 19*a^2*b^4*c^8 + b^6*c^8 - 3*a^4*c^10 - 25*a^2*b^2*c^10 - 5*b^4*c^10 + 5*a^2*c^12 + 5*b^2*c^12 - c^14) : :

(14656) lies on the tangential circle and these lines: {3, 126}, {22, 1296}, {24, 111}, {25, 5512}, {378, 10734}, {543, 14070}, {2070, 11258}, {2780, 10117}, {2854, 2931}, {3048, 9707}, {5926, 7669}, {6642, 6719}, {7488, 14360}, {9683, 11835}, {9694, 11833}

X(14658) =  SINGULAR FOCUS OF THE CUBIC K249

Barycentrics    a^2*(a^6 + 2*a^4*b^2 + 2*a^2*b^4 + b^6 - 7*a^4*c^2 - 7*b^4*c^2 + 5*a^2*c^4 + 5*b^2*c^4 - 2*c^6)*(a^6 - 7*a^4*b^2 + 5*a^2*b^4 - 2*b^6 + 2*a^4*c^2 + 5*b^4*c^2 + 2*a^2*c^4 - 7*b^2*c^4 + c^6) : :

X(14658) lies on the circumcircle, the cubic K249, and these lines: {1291, 1627}, {2709, 5097}

X(14658) = circumcircle intercept, other than X(930), of circle {{X(2),X(17),X(18)}}
X(14658) = circumcircle intercept, orther than X(110), of circle {{X(2),X(61),X(62)}}
X(14658) = intercept, other than X(2), of circles {{X(2),X(17),X(18)}} and {{X(2),X(61),X(62)}}

X(14659) =  SINGULAR FOCUS OF THE CUBIC K250

Barycentrics    a^2*(a^6 + a^4*b^2 + a^2*b^4 + b^6 - 5*a^4*c^2 - 5*b^4*c^2 + 4*a^2*c^4 + 4*b^2*c^4 - 2*c^6)*(a^6 - 5*a^4*b^2 + 4*a^2*b^4 - 2*b^6 + a^4*c^2 + 4*b^4*c^2 + a^2*c^4 - 5*b^2*c^4 + c^6) : :

X(14659) lies on the circumcircle, the cubic K250, and these lines: {2, 2858}, {6, 10425}, {99, 230}, {110, 1692}, {111, 8651}, {112, 5140}, {187, 3565}, {691, 3053}, {1351, 2709}, {2374, 2501}, {2489, 3563}, {2855, 7735}, {6082, 6792}

X(14659) = isogonal conjugate of X(14645)
X(14659) = circumcircle intercept, other than X(110), of circle {{X(2),X(371),X(372)}}
X(14659) = circumcircle intercept, other than X(925), of circle {{X(2),X(485),X(486)}}
X(14659) = intercept, other than X(2), of circles {{X(2),X(371),X(372)}} and {{X(2),X(485),X(486)}}
X(14659) = Schoutte-circle inverse of X(3565)
X(14659) = trilinear pole of line {6, 2872}

X(14660) =  SINGULAR FOCUS OF THE CUBIC K291

Barycentrics    a^2*(a^8 - a^6*b^2 - a^2*b^6 - b^8 - a^6*c^2 + 2*a^2*b^4*c^2 + 2*a^2*b^2*c^4 + b^4*c^4 - a^2*c^6 - c^8) : :

X(14660) lies on the Parry circle and these lines: {2, 4048}, {3, 9998}, {6, 9999}, {23, 1691}, {39, 110}, {51, 251}, {111, 5092}, {694, 6636}, {1180, 3506}, {1640, 9147}, {2001, 5028}, {2502, 7711}, {5027, 14403}, {5162, 8569}, {11205, 14153}
{X(2502),X(12055)}-harmonic conjugate of X(7711)

X(14660) = circumcircle-inverse of X(9998)
X(14660) = Parry-isodynamic-circle inverse of X(7711)
X(14660) = Psi-transform of X(32)
X(14660) = crossdifference of every pair of points on line {5113, 14420}

X(14661) =  SINGULAR FOCUS OF THE CUBIC K299

Barycentrics    a*(a^7 - 2*a^6*b - a^5*b^2 + 4*a^4*b^3 - a^3*b^4 - 2*a^2*b^5 + a*b^6 - 2*a^6*c + 5*a^5*b*c - a^4*b^2*c - a^3*b^3*c - 3*a^2*b^4*c + 4*a*b^5*c - 2*b^6*c - a^5*c^2 - a^4*b*c^2 - 6*a^3*b^2*c^2 + 7*a^2*b^3*c^2 - a*b^4*c^2 + 2*b^5*c^2 + 4*a^4*c^3 - a^3*b*c^3 + 7*a^2*b^2*c^3 - 8*a*b^3*c^3 - a^3*c^4 - 3*a^2*b*c^4 - a*b^2*c^4 - 2*a^2*c^5 + 4*a*b*c^5 + 2*b^2*c^5 + a*c^6 - 2*b*c^6) : :

X(14661) lies on these lines: {3, 667}, {220, 1566}, {518, 1351}, {5526, 5587}, {5687, 6065}

X(14661) = reflection of X(3) in X(1083)

X(14662) =  SINGULAR FOCUS OF THE CUBIC K304

Barycentrics    a^2*(a^8 - 2*a^6*b^2 + 2*a^4*b^4 + 2*a^2*b^6 - 3*b^8 - 2*a^6*c^2 - a^4*b^2*c^2 - a^2*b^4*c^2 + 13*b^6*c^2 + 2*a^4*c^4 - a^2*b^2*c^4 - 22*b^4*c^4 + 2*a^2*c^6 + 13*b^2*c^6 - 3*c^8) : :

X(14662) lies on these lines: {23, 111}, {114, 381}, {576, 2854}, {1296, 5966}

X(14663) =  SINGULAR FOCUS OF THE CUBIC K306

Barycentrics    a*(a^9 - a^8*b - 2*a^7*b^2 + 3*a^6*b^3 - 3*a^4*b^5 + 2*a^3*b^6 + a^2*b^7 - a*b^8 - a^8*c + 4*a^7*b*c - 3*a^6*b^2*c - 4*a^5*b^3*c + 8*a^4*b^4*c - 3*a^3*b^5*c - 2*a^2*b^6*c + 3*a*b^7*c - 2*b^8*c - 2*a^7*c^2 - 3*a^6*b*c^2 + 5*a^5*b^2*c^2 - 2*a^4*b^3*c^2 - 3*a^3*b^4*c^2 + 3*a^2*b^5*c^2 + 2*b^7*c^2 + 3*a^6*c^3 - 4*a^5*b*c^3 - 2*a^4*b^2*c^3 + 10*a^3*b^3*c^3 - 4*a^2*b^4*c^3 - 3*a*b^5*c^3 + 6*b^6*c^3 + 8*a^4*b*c^4 - 3*a^3*b^2*c^4 - 4*a^2*b^3*c^4 + 2*a*b^4*c^4 - 6*b^5*c^4 - 3*a^4*c^5 - 3*a^3*b*c^5 + 3*a^2*b^2*c^5 - 3*a*b^3*c^5 - 6*b^4*c^5 + 2*a^3*c^6 - 2*a^2*b*c^6 + 6*b^3*c^6 + a^2*c^7 + 3*a*b*c^7 + 2*b^2*c^7 - a*c^8 - 2*b*c^8) : :
X(14663) = 3 X(3) - 2 X(6011) = 3 X(759) - X(6011)

X(14663) lies on the Stammler circle and these lines: {3, 759}, {355, 8715}, {399, 1482}, {999, 1365}, {6044, 9567}

X(14663) = reflection of X(3) in X(759)

X(14664) =  SINGULAR FOCUS OF THE CUBIC K338

Barycentrics    a*(2*a^5 - 5*a^4*b - 2*a^3*b^2 + 6*a^2*b^3 - b^5 - 5*a^4*c + 22*a^3*b*c - 13*a^2*b^2*c - 8*a*b^3*c + 4*b^4*c - 2*a^3*c^2 - 13*a^2*b*c^2 + 20*a*b^2*c^2 - 3*b^3*c^2 + 6*a^2*c^3 - 8*a*b*c^3 - 3*b^2*c^3 + 4*b*c^4 - c^5) : :
X(14664) = 3 X(165) + X(1054) = 3 X(165) - X(1293) = 3 X(3576) - X(10700) = 3 X(5587) - X(10730) = 3 X(10165) - 2 X(11731) = 3 X(10164) - X(11814) = 5 X(7987) - X(13541)

X(14664) lies on these lines: {3, 2802}, {40, 106}, {105, 165}, {121, 6684}, {516, 5510}, {517, 11717}, {659, 2827}, {946, 6715}, {1155, 1357}, {2796, 6055}, {3038, 4640}, {3576, 10700}, {5587, 10730}, {7987, 13541}, {10164, 11814}, {10165, 11731}

X(14664) = midpoint of X(i) and X(j) for these {i,j}: {40, 106}, {1054, 1293}
X(14664) = reflection of X(i) in X(j) for these {i,j}: {121, 6684}, {946, 6715}
X(14664) = circumcircle inverse of X(13205)
X(14664) = X(2827)-vertex conjugate of X(13205)
X(14664) = {X(165),X(1054)}-harmonic conjugate of X(1293)

X(14665) =  SINGULAR FOCUS OF THE CUBIC K359

Barycentrics    a*(-2*a^2*b^2 + a^3*c + a^2*b*c + a*b^2*c + b^3*c - a^2*c^2 - b^2*c^2)*(a^3*b - a^2*b^2 + a^2*b*c - 2*a^2*c^2 + a*b*c^2 - b^2*c^2 + b*c^3) : :

X(14665) lies on the circumcircle, the cubic K359, and these lines: {1, 813}, {32, 919}, {56, 927}, {99, 3286}, {100, 239}, {101, 238}, {105, 667}, {109, 1429}, {182, 2742}, {741, 1019}, {805, 3110}, {898, 1001}, {901, 5091}, {1016, 8299}, {1078, 8709}, {1083, 6078}, {1293, 9441}, {1385, 2704}, {1386, 2703}, {2702, 12194}, {3230, 8693}, {6551, 13194}, {8684, 12195}

X(14665) = X(21)-beth conjugate of X(813)
X(14665) = trilinear pole of line {6, 659}

X(14666) =  SINGULAR FOCUS OF THE CUBIC K394

Barycentrics    5*a^10 - 16*a^8*b^2 - a^6*b^4 + 17*a^4*b^6 - 4*a^2*b^8 - b^10 - 16*a^8*c^2 + 59*a^6*b^2*c^2 - 36*a^4*b^4*c^2 + 8*a^2*b^6*c^2 + 5*b^8*c^2 - a^6*c^4 - 36*a^4*b^2*c^4 - 4*b^6*c^4 + 17*a^4*c^6 + 8*a^2*b^2*c^6 - 4*b^4*c^6 - 4*a^2*c^8 + 5*b^2*c^8 - c^10 : :

X(14666) lies on these lines: {2, 10748}, {3, 543}, {30, 111}, {126, 5054}, {381, 9172}, {549, 10717}, {1296, 8703}, {2080, 6094}, {2854, 11179}, {3524, 14360}, {3534, 11258}, {3656, 11721}, {3830, 5512}, {3845, 10734}, {5055, 6719}, {5655, 9129}, {11178, 14605}

X(14666) = midpoint of X(3534) and X(11258)
X(14666) = reflection of X(i) in X(j) for these {i,j}: {381, 9172}, {1296, 8703}, {3656, 11721}, {3830, 5512}, {5655, 9129}, {10717, 549}, {10734, 3845}, {10748, 2}
X(14666) = circumcircle-inverse of X(13233)
X(14666) = X(2793)-vertex conjugate of X(13233)

X(14667) =  SINGULAR FOCUS OF THE CUBIC K436

Barycentrics    a^2*(a^7 - a^6*b - a^5*b^2 + a^4*b^3 - a^3*b^4 + a^2*b^5 + a*b^6 - b^7 - a^6*c + a^5*b*c + a^4*b^2*c - a^2*b^4*c - a*b^5*c + b^6*c - a^5*c^2 + a^4*b*c^2 - a*b^4*c^2 + b^5*c^2 + a^4*c^3 + 2*a*b^3*c^3 - b^4*c^3 - a^3*c^4 - a^2*b*c^4 - a*b^2*c^4 - b^3*c^4 + a^2*c^5 - a*b*c^5 + b^2*c^5 + a*c^6 + b*c^6 - c^7) : :

X(14667) lies on the tangential circle and these lines: {1, 2931}, {3, 11}, {22, 105}, {24, 108}, {25, 5521}, {26, 7742}, {36, 1421}, {55, 11028}, {56, 11700}, {405, 9659}, {1360, 1617}, {2006, 10260}, {2078, 9590}, {2218, 2915}, {5197, 7295}, {6129, 10016}, {6644, 8069}, {7163, 8757}, {7580, 9673}, {10776, 14127}, {11713, 12740}

X(14667) = circumcircle-inverse of X(11)

X(14668) =  SINGULAR FOCUS OF THE CUBIC K439

Barycentrics    a^2*(a^16 - 5*a^14*b^2 + 11*a^12*b^4 - 13*a^10*b^6 + 5*a^8*b^8 + 9*a^6*b^10 - 15*a^4*b^12 + 9*a^2*b^14 - 2*b^16 - 5*a^14*c^2 + 13*a^12*b^2*c^2 - 9*a^10*b^4*c^2 - 5*a^8*b^6*c^2 + a^6*b^8*c^2 + 27*a^4*b^10*c^2 - 35*a^2*b^12*c^2 + 13*b^14*c^2 + 11*a^12*c^4 - 9*a^10*b^2*c^4 - 9*a^8*b^4*c^4 + 17*a^6*b^6*c^4 - 29*a^4*b^8*c^4 + 61*a^2*b^10*c^4 - 42*b^12*c^4 - 13*a^10*c^6 - 5*a^8*b^2*c^6 + 17*a^6*b^4*c^6 + 7*a^4*b^6*c^6 - 35*a^2*b^8*c^6 + 83*b^10*c^6 + 5*a^8*c^8 + a^6*b^2*c^8 - 29*a^4*b^4*c^8 - 35*a^2*b^6*c^8 - 104*b^8*c^8 + 9*a^6*c^10 + 27*a^4*b^2*c^10 + 61*a^2*b^4*c^10 + 83*b^6*c^10 - 15*a^4*c^12 - 35*a^2*b^2*c^12 - 42*b^4*c^12 + 9*a^2*c^14 + 13*b^2*c^14 - 2*c^16) : :

X(14668) lies on these lines: {5, 930}, {1157, 5899}, {5966, 6636}, {13564, 14367}

X(14669) =  SINGULAR FOCUS OF THE CUBIC K441

Barycentrics    a^2*(a^12 - 3*a^10*b^2 + 4*a^8*b^4 - 2*a^6*b^6 - 3*a^4*b^8 + 5*a^2*b^10 - 2*b^12 - 3*a^10*c^2 + a^8*b^2*c^2 + 4*a^6*b^4*c^2 - 2*a^4*b^6*c^2 - 9*a^2*b^8*c^2 + 9*b^10*c^2 + 4*a^8*c^4 + 4*a^6*b^2*c^4 - 6*a^4*b^4*c^4 + 8*a^2*b^6*c^4 - 22*b^8*c^4 - 2*a^6*c^6 - 2*a^4*b^2*c^6 + 8*a^2*b^4*c^6 + 30*b^6*c^6 - 3*a^4*c^8 - 9*a^2*b^2*c^8 - 22*b^4*c^8 + 5*a^2*c^10 + 9*b^2*c^10 - 2*c^12) : :

X(14669) lies on lines: {4, 99}, {26, 2079}, {155, 2930}, {3565, 5966}, {5104, 11675}, {6091, 7556}, {7517, 11641}

X(14669) = circumcircle-inverse of X(15560)
X(14669) = singular focus of the cubic K441

X(14670) =  SINGULAR FOCUS OF THE CUBIC K448

Barycentrics    a^2*(a^10*b^4 - 5*a^8*b^6 + 10*a^6*b^8 - 10*a^4*b^10 + 5*a^2*b^12 - b^14 + 2*a^8*b^4*c^2 - 7*a^6*b^6*c^2 + 9*a^4*b^8*c^2 - 5*a^2*b^10*c^2 + b^12*c^2 + a^10*c^4 + 2*a^8*b^2*c^4 - 6*a^2*b^8*c^4 + 3*b^10*c^4 - 5*a^8*c^6 - 7*a^6*b^2*c^6 + 12*a^2*b^6*c^6 - 3*b^8*c^6 + 10*a^6*c^8 + 9*a^4*b^2*c^8 - 6*a^2*b^4*c^8 - 3*b^6*c^8 - 10*a^4*c^10 - 5*a^2*b^2*c^10 + 3*b^4*c^10 + 5*a^2*c^12 + b^2*c^12 - c^14) : :

X(14670) lies on these lines: {3, 74}, {5, 523}

X(14670) = midpoint of X(3) and X(14264)
X(14670) = reflection of X(14254) in X(5)
X(14670) = crossdifference of every pair of points on line {50, 1637}
X(14670) = X(14254)-of-Johnson-triangle
X(14670) = {X(10287),X(10288)}-harmonic conjugate of X(5)

X(14671) =  SINGULAR FOCUS OF THE CUBIC K468

Barycentrics    a^2*(a^12 - 5*a^10*b^2 + 7*a^8*b^4 - 7*a^4*b^8 + 5*a^2*b^10 - b^12 - 5*a^10*c^2 + 18*a^8*b^2*c^2 - 19*a^6*b^4*c^2 + 16*a^4*b^6*c^2 - 16*a^2*b^8*c^2 + 6*b^10*c^2 + 7*a^8*c^4 - 19*a^6*b^2*c^4 - 3*a^4*b^4*c^4 + 11*a^2*b^6*c^4 - 14*b^8*c^4 + 16*a^4*b^2*c^6 + 11*a^2*b^4*c^6 + 18*b^6*c^6 - 7*a^4*c^8 - 16*a^2*b^2*c^8 - 14*b^4*c^8 + 5*a^2*c^10 + 6*b^2*c^10 - c^12) : :

X(14671) lies on these lines: {3, 148}, {54, 575}, {112, 3199}

X(14672) =  SINGULAR FOCUS OF THE CUBIC K475

Barycentrics    (b - c)^2*(b + c)^2*(-a^2 + b^2 + c^2)*(-a^4 + b^4 - 4*b^2*c^2 + c^4)*(3*a^6 - a^4*b^2 - 3*a^2*b^4 + b^6 - a^4*c^2 + 2*a^2*b^2*c^2 - b^4*c^2 - 3*a^2*c^4 - b^2*c^4 + c^6) : :

X(14672) lies on the nine-point circle and these lines: {3, 126}, {4, 2373}, {5, 1560}, {113, 1352}, {132, 381}, {136, 3143}, {339, 5139}, {10297, 13449}

X(14672) = midpoint of X(4) and X(2373)
X(14672) = reflection of X(1560) in X(5)
X(14672) = X(1560)-of-Johnson-triangle
X(14672) = polar-circle-inverse of X(39382)
X(14672) = X(i)-complementary conjugate of X(j) for these (i,j): {810, 574}, {1995, 8062}, {14209, 141}

X(14673) =  SINGULAR FOCUS OF THE CUBIC K523

Barycentrics    a^2*(a^10 - 5*a^6*b^4 + 5*a^4*b^6 - b^10 + 7*a^6*b^2*c^2 - 4*a^4*b^4*c^2 - 5*a^2*b^6*c^2 + 2*b^8*c^2 - 5*a^6*c^4 - 4*a^4*b^2*c^4 + 10*a^2*b^4*c^4 - b^6*c^4 + 5*a^4*c^6 - 5*a^2*b^2*c^6 - b^4*c^6 + 2*b^2*c^8 - c^10) : :

X(14673) lies on these lines: {3, 113}, {24, 5667}, {25, 98}, {133, 1598}, {1294, 11414}, {1593, 10152}, {1650, 13203}, {1661, 9530}, {2797, 13175}, {2803, 13222}, {2828, 9913}, {2848, 11641}, {5020, 6716}, {7158, 10833}, {8192, 10701}, {9033, 12310}, {11365, 11718}, {13507, 13621}

X(14673) = circumcircle-inverse of X(5972)
X(14673) = Stammler-circle-inverse of X(7728)
X(14673) = X(5972)-vertex conjugate of X(9033)
X(14673) = circumperp conjugate of X(37853)
X(14673) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1624, 10117, 3)

X(14674) =  SINGULAR FOCUS OF THE CUBIC K563

Barycentrics    a^16 - 4*a^14*b^2 + 6*a^12*b^4 - 5*a^10*b^6 + 5*a^8*b^8 - 6*a^6*b^10 + 4*a^4*b^12 - a^2*b^14 - 4*a^14*c^2 + 12*a^12*b^2*c^2 - 12*a^10*b^4*c^2 + 2*a^8*b^6*c^2 + 7*a^6*b^8*c^2 - 9*a^4*b^10*c^2 + 5*a^2*b^12*c^2 - b^14*c^2 + 6*a^12*c^4 - 12*a^10*b^2*c^4 + 9*a^8*b^4*c^4 - 4*a^6*b^6*c^4 + 4*a^4*b^8*c^4 - 9*a^2*b^10*c^4 + 6*b^12*c^4 - 5*a^10*c^6 + 2*a^8*b^2*c^6 - 4*a^6*b^4*c^6 + 2*a^4*b^6*c^6 + 5*a^2*b^8*c^6 - 15*b^10*c^6 + 5*a^8*c^8 + 7*a^6*b^2*c^8 + 4*a^4*b^4*c^8 + 5*a^2*b^6*c^8 + 20*b^8*c^8 - 6*a^6*c^10 - 9*a^4*b^2*c^10 - 9*a^2*b^4*c^10 - 15*b^6*c^10 + 4*a^4*c^12 + 5*a^2*b^2*c^12 + 6*b^4*c^12 - a^2*c^14 - b^2*c^14 : :

X(14674) lies on thesae lines: {5, 49}, {137, 13557}, {186, 2970}, {930, 13530}

X(14674) = U-inverse of (49), where U = the circle with diameter X(3)X(4)

X(14675) =  SINGULAR FOCUS OF THE CUBIC K629

Barycentrics    a^2*(a^12 - 2*a^10*b^2 + 2*a^6*b^6 - a^4*b^8 - 2*a^10*c^2 + a^6*b^4*c^2 + 2*a^4*b^6*c^2 - b^10*c^2 + a^6*b^2*c^4 - 3*a^4*b^4*c^4 - a^2*b^6*c^4 + 2*b^8*c^4 + 2*a^6*c^6 + 2*a^4*b^2*c^6 - a^2*b^4*c^6 - 2*b^6*c^6 - a^4*c^8 + 2*b^4*c^8 - b^2*c^10) : :

X(14576) lies on these lines: {5, 83}, {32, 2079}, {182, 5621}, {186, 2080}, {1078, 10125}, {6240, 11380}, {11561, 13193}, {12054, 14118}

X(14676) =  SINGULAR FOCUS OF THE CUBIC K630

Barycentrics    a^2*(a^12 - 3*a^10*b^2 + 2*a^8*b^4 + 2*a^6*b^6 - 3*a^4*b^8 + a^2*b^10 - 3*a^10*c^2 + 3*a^8*b^2*c^2 - 2*a^6*b^4*c^2 + 3*a^4*b^6*c^2 - a^2*b^8*c^2 + 2*a^8*c^4 - 2*a^6*b^2*c^4 - 2*a^4*b^4*c^4 + 2*b^8*c^4 + 2*a^6*c^6 + 3*a^4*b^2*c^6 - 4*b^6*c^6 - 3*a^4*c^8 - a^2*b^2*c^8 + 2*b^4*c^8 + a^2*c^10) : :

X(14676) lies on these lines: {4, 32}, {127, 6639}, {182, 2781}, {1078, 13219}, {1297, 8722}, {1658, 5171}, {2070, 2080}, {3398, 14130}, {5034, 10766}, {6720, 7808}, {10104, 13406}, {10254, 10749}, {10800, 11722}, {11364, 13221}

X(14677) =  SINGULAR FOCUS OF THE CUBIC K648

Barycentrics    4*a^10 - 5*a^8*b^2 - 8*a^6*b^4 + 14*a^4*b^6 - 4*a^2*b^8 - b^10 - 5*a^8*c^2 + 24*a^6*b^2*c^2 - 15*a^4*b^4*c^2 - 7*a^2*b^6*c^2 + 3*b^8*c^2 - 8*a^6*c^4 - 15*a^4*b^2*c^4 + 22*a^2*b^4*c^4 - 2*b^6*c^4 + 14*a^4*c^6 - 7*a^2*b^2*c^6 - 2*b^4*c^6 - 4*a^2*c^8 + 3*b^2*c^8 - c^10 : :
X(14677) = 3 X(3) - X(146) = 3 X(74) - X(265) = 3 X(376) - X(399) = 2 X(113) - 3 X(549) = 3 X(5) - 2 X(1539) = 3 X(1511) - 2 X(6053) = 3 X(5) - 4 X(6699) = 7 X(1539) - 12 X(6723) = 7 X(5) - 8 X(6723) = 7 X(6699) - 6 X(6723) = 4 X(6053) - 9 X(8703) = 2 X(1511) - 3 X(8703) = 7 X(265) - 9 X(9140) = 7 X(74) - 3 X(9140) = 6 X(9140) - 7 X(10264) = 2 X(265) - 3 X(10264) = 3 X(3) - 2 X(10272) = 15 X(9140) - 7 X(10733) = 5 X(265) - 3 X(10733) = 5 X(10264) - 2 X(10733) = 5 X(74) - X(10733) = X(550) + 2 X(10990) = 3 X(10304) - 2 X(11694) = 3 X(5946) - 2 X(11807) = 4 X(6723) - 7 X(12041) = X(1539) - 3 X(12041) = 2 X(6699) - 3 X(12041) = X(146) + 3 X(12244) = 2 X(10272) + 3 X(12244) = 3 X(10620) - X(12317) = 3 X(20) + X(12317) = 3 X(3534) - X(12383) = 9 X(11539) - 8 X(12900) = 3 X(3845) - 2 X(13202) = X(12273) - 3 X(13340) = 3 X(5655) - 4 X(13392)

X(14677) lies on these lines: {3, 146}, {5, 1539}, {20, 10620}, {30, 74}, {110, 548}, {113, 549}, {125, 3627}, {140, 7728}, {376, 399}, {382, 11801}, {541, 1511}, {542, 3630}, {546, 10721}, {550, 5562}, {1597, 11566}, {1657, 3448}, {2771, 4067}, {2935, 4846}, {3529, 12902}, {3534, 12383}, {3845, 13202}, {5655, 13392}, {5900, 11559}, {5946, 11807}, {6644, 9919}, {10263, 11806}, {10304, 11694}, {10628, 13491}, {10706, 12100}, {11468, 13406}, {11539, 12900}, {12084, 13171}, {12103, 12121}, {12273, 13340}, {13417, 13630}

X(14677) = midpoint of X(i) and X(j) for these {i,j}: {3, 12244}, {20, 10620}, {1657, 3448}, {3529, 12902}
X(14677) = reflection of X(i) in X(j) for these {i,j}: {5, 12041}, {110, 548}, {146, 10272}, {382, 11801}, {1539, 6699}, {3627, 125}, {7728, 140}, {10263, 11806}, {10264, 74}, {10706, 12100}, {10721, 546}, {12121, 12103}, {13417, 13630}
X(14677) = complement of X(38790)
X(14677) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 146, 10272), (1539, 6699, 5), (1539, 12041, 6699)

X(14678) =  SINGULAR FOCUS OF THE CUBIC K653

Barycentrics    a^2*(-2*a^10*b^6 + 2*a^6*b^10 + a^12*b^2*c^2 - 2*a^8*b^6*c^2 + 5*a^6*b^8*c^2 + 2*a^4*b^10*c^2 - 2*a^2*b^12*c^2 + 2*a^6*b^6*c^4 + a^2*b^10*c^4 - 2*a^10*c^6 - 2*a^8*b^2*c^6 + 2*a^6*b^4*c^6 - 9*a^4*b^6*c^6 - 6*a^2*b^8*c^6 + 4*b^10*c^6 + 5*a^6*b^2*c^8 - 6*a^2*b^6*c^8 + 2*a^6*c^10 + 2*a^4*b^2*c^10 + a^2*b^4*c^10 + 4*b^6*c^10 - 2*a^2*b^2*c^12) : :

X(14678) lies on these lines: {3, 8150}, {111, 7467}, {733, 6234}

X(14678) = circumcircle-inverse of X(8178)

X(14679) =  SINGULAR FOCUS OF THE CUBIC K536

Barycentrics    a^10 - a^9*b - 2*a^8*b^2 + 3*a^7*b^3 - 3*a^5*b^5 + 2*a^4*b^6 + a^3*b^7 - a^2*b^8 - a^9*c + 3*a^8*b*c - a^7*b^2*c - 4*a^6*b^3*c + 5*a^5*b^4*c - 2*a^4*b^5*c - 3*a^3*b^6*c + 4*a^2*b^7*c - b^9*c - 2*a^8*c^2 - a^7*b*c^2 + 6*a^6*b^2*c^2 - 2*a^5*b^3*c^2 - 2*a^4*b^4*c^2 + 3*a^3*b^5*c^2 - 2*a^2*b^6*c^2 + 3*a^7*c^3 - 4*a^6*b*c^3 - 2*a^5*b^2*c^3 + 4*a^4*b^3*c^3 - a^3*b^4*c^3 - 4*a^2*b^5*c^3 + 4*b^7*c^3 + 5*a^5*b*c^4 - 2*a^4*b^2*c^4 - a^3*b^3*c^4 + 6*a^2*b^4*c^4 - 3*a^5*c^5 - 2*a^4*b*c^5 + 3*a^3*b^2*c^5 - 4*a^2*b^3*c^5 - 6*b^5*c^5 + 2*a^4*c^6 - 3*a^3*b*c^6 - 2*a^2*b^2*c^6 + a^3*c^7 + 4*a^2*b*c^7 + 4*b^3*c^7 - a^2*c^8 - b*c^9 : :

X(14679) lies on these lines: {1, 5}, {3, 4858}, {28, 104}, {100, 7549}, {153, 7557}, {1715, 12515}, {1871, 2831}, {2829, 7511}, {3120, 10747}, {6713, 7561}, {7562, 12773}

X(14680) =  SINGULAR FOCUS OF THE CUBIC K720

Barycentrics    a^8*b - 4*a^6*b^3 + 2*a^5*b^4 + 4*a^4*b^5 - 4*a^3*b^6 + 2*a*b^8 - b^9 + a^8*c - 4*a^7*b*c + 2*a^6*b^2*c + 4*a^5*b^3*c - 6*a^4*b^4*c + 2*a^3*b^5*c + 2*a^2*b^6*c - 2*a*b^7*c + b^8*c + 2*a^6*b*c^2 - 4*a^5*b^2*c^2 + 3*a^4*b^3*c^2 + 4*a^3*b^4*c^2 - 5*a^2*b^5*c^2 - 2*a*b^6*c^2 + 2*b^7*c^2 - 4*a^6*c^3 + 4*a^5*b*c^3 + 3*a^4*b^2*c^3 - 8*a^3*b^3*c^3 + 3*a^2*b^4*c^3 + 2*a*b^5*c^3 - 2*b^6*c^3 + 2*a^5*c^4 - 6*a^4*b*c^4 + 4*a^3*b^2*c^4 + 3*a^2*b^3*c^4 + 4*a^4*c^5 + 2*a^3*b*c^5 - 5*a^2*b^2*c^5 + 2*a*b^3*c^5 - 4*a^3*c^6 + 2*a^2*b*c^6 - 2*a*b^2*c^6 - 2*b^3*c^6 - 2*a*b*c^7 + 2*b^2*c^7 + 2*a*c^8 + b*c^8 - c^9 : :

X(14680) lies on these lines: {5, 5620}, {355, 8715}, {952, 13604}, {2752, 6011}

X(14680) = reflection of X(5620) in X(5)
X(14680) = X(5620)-of-Johnson-triangle

X(14681) =  SINGULAR FOCUS OF THE CUBIC K728

Barycentrics    a^2*(4*a^12 - 4*a^10*b^2 - 2*a^8*b^4 - 4*a^6*b^6 + 4*a^4*b^8 + 8*a^2*b^10 - 6*b^12 - 4*a^10*c^2 - 13*a^8*b^2*c^2 + 20*a^6*b^4*c^2 - 9*a^4*b^6*c^2 - a^2*b^8*c^2 + 7*b^10*c^2 - 2*a^8*c^4 + 20*a^6*b^2*c^4 + 9*a^4*b^4*c^4 - 12*a^2*b^6*c^4 - 12*b^8*c^4 - 4*a^6*c^6 - 9*a^4*b^2*c^6 - 12*a^2*b^4*c^6 + 22*b^6*c^6 + 4*a^4*c^8 - a^2*b^2*c^8 - 12*b^4*c^8 + 8*a^2*c^10 + 7*b^2*c^10 - 6*c^12) : :

X(14681) lies on these lines: {147, 376}, {399, 1350}, {842, 2080}, {9821, 9999}, {12074, 14388}

X(14682) =  SINGULAR FOCUS OF THE CUBIC K730

Barycentrics    a^2*(4*a^10 - 6*a^8*b^2 - 10*a^6*b^4 + 10*a^4*b^6 + 6*a^2*b^8 - 4*b^10 - 6*a^8*c^2 + 13*a^6*b^2*c^2 + 3*a^4*b^4*c^2 - 12*a^2*b^6*c^2 + 4*b^8*c^2 - 10*a^6*c^4 + 3*a^4*b^2*c^4 + 9*a^2*b^4*c^4 - 4*b^6*c^4 + 10*a^4*c^6 - 12*a^2*b^2*c^6 - 4*b^4*c^6 + 6*a^2*c^8 + 4*b^2*c^8 - 4*c^10) : :

X(14682) lies on these lines: {2, 11643}, {3, 9829}, {22, 2930}, {23, 3849}, {99, 5987}, {110, 5104}, {1995, 2079}, {2770, 6236}, {7496, 10163}

X(14682) = reflection of X(6325) in X(6031)
X(14682) =circumcircle-inverse of X(10717)
X(14682) = 1st-Ehrmann-to-circummedial similarity image of X(12584)

X(14683) = SINGULAR FOCUS OF THE CUBIC K753

Barycentrics    3*a^6 - 3*a^4*b^2 + a^2*b^4 - b^6 - 3*a^4*c^2 + a^2*b^2*c^2 + b^4*c^2 + a^2*c^4 + b^2*c^4 - c^6 : :
X(14683) = 3 X(2) - 4 X(110) = 9 X(2) - 8 X(125) = 3 X(110) - 2 X(125) = 4 X(265) - 5 X(3091) = 4 X(125) - 3 X(3448) = 4 X(74) - 5 X(3522) = 8 X(1511) - 7 X(3523) = 4 X(67) - 5 X(3620) = 8 X(113) - 7 X(3832) = 5 X(3091) - 8 X(5609) = 7 X(3448) - 12 X(5642) = 7 X(125) - 9 X(5642) = 7 X(2) - 8 X(5642) = 7 X(110) - 6 X(5642) = 3 X(3839) - 4 X(5655) = 15 X(2) - 16 X(5972) = 15 X(5642) - 14 X(5972) = 5 X(3448) - 8 X(5972) = 5 X(125) - 6 X(5972) = 5 X(110) - 4 X(5972) = 11 X(3448) - 16 X(6723) = 11 X(125) - 12 X(6723) = 11 X(5972) - 10 X(6723) = 11 X(110) - 8 X(6723) = 17 X(3854) - 16 X(7687) = 3 X(3543) - 4 X(7728) = 5 X(3623) - 4 X(7984) = 10 X(125) - 9 X(9140) = 10 X(5642) - 7 X(9140) = 5 X(3448) - 6 X(9140) = 5 X(2) - 4 X(9140) = 5 X(110) - 3 X(9140) = 4 X(5972) - 3 X(9140) = 8 X(5972) - 15 X(9143) = 4 X(125) - 9 X(9143) = 4 X(5642) - 7 X(9143) = 2 X(9140) - 5 X(9143) = 2 X(110) - 3 X(9143) = X(3448) - 3 X(9143) = 3 X(9778) - 2 X(9904) = 9 X(3839) - 8 X(10113) = 3 X(5655) - 2 X(10113) = 5 X(631) - 4 X(10264) = 7 X(3090) - 8 X(10272) = 3 X(376) - 2 X(10620) = 3 X(3146) - 4 X(10721)

Let T the Steiner triangle, E the Euler line of ABC, and P the parabola inscribed in T with directrix E. The focus of P is X(14683). (Angel Montesdeoca, February 14, 2020)

X(14683) lies on the circumcircle of the Steiner triangle and on these lines: {2, 98}, {3, 5900}, {4, 195}, {6, 7533}, {8, 2948}, {20, 5663}, {22, 11898}, {23, 3564}, {30, 12308}, {67, 3620}, {69, 2916}, {74, 3522}, {113, 3832}, {146, 3146}, {155, 12319}, {193, 2854}, {246, 9862}, {265, 3091}, {323, 1503}, {376, 10620}, {390, 3024}, {539, 14157}, {575, 7605}, {631, 10264}, {944, 2771}, {1112, 6995}, {1511, 3523}, {1587, 12375}, {1588, 12376}, {1614, 2888}, {1986, 7487}, {1994, 5480}, {2777, 5059}, {2836, 4430}, {2931, 11411}, {2935, 12324}, {2937, 5898}, {3028, 3600}, {3043, 3541}, {3090, 10272}, {3474, 11670}, {3526, 13392}, {3543, 7728}, {3545, 11801}, {3575, 12165}, {3616, 13605}, {3617, 13211}, {3622, 11720}, {3623, 7984}, {3818, 11422}, {3839, 5655}, {3854, 7687}, {4294, 7727}, {4549, 12281}, {4563, 14360}, {5261, 12903}, {5274, 12904}, {5334, 10658}, {5335, 10657}, {5603, 11699}, {5876, 12254}, {6636, 13171}, {7404, 12228}, {7488, 12412}, {7731, 9936}, {9778, 9904}, {9833, 10628}, {9919, 12087}, {10117, 11206}, {10263, 11271}, {10304, 12041}, {10519, 12584}, {10706, 12295}, {11002, 11800}, {11424, 14049}, {11562, 12284}, {12168, 14118}

X(14683) = anticomplement X(3448)
X(14683) = reflection of X(i) in X(j) for these {i,j}: {2, 9143}, {4, 399}, {8, 2948}, {20, 12383}, {69, 2930}, {146, 14094}, {193, 11061}, {265, 5609}, {3146, 146}, {3448, 110}, {5189, 323}, {11411, 2931}, {12244, 12121}, {12284, 11562}, {12317, 3}, {12319, 155}, {12324, 2935}, {12325, 5898}
X(14683) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {3447, 8}, {13485, 6327}
X(14683) = X(13485)-Ceva conjugate of X(2)
X(14683) = orthoptic-circle-of-Steiner-inellipe-inverse of X(6721)
X(14683) = orthoptic-circle-of-Steiner-circumellipe-inverse of X(114)
X(14683) = Psi-transform of X(140)
X(14683) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (110, 3448, 2), (110, 9140, 5972), (184, 3410, 2), (1352, 11003, 2), (3448, 9143, 110), (9544, 11442, 2), (12121, 12244, 20), (12244, 12383, 12121)

X(14684) =  SINGULAR FOCUS OF THE CUBIC K793

Barycentrics    a^2*(4*a^8*b^4 - 2*a^6*b^6 - 4*a^4*b^8 + 2*a^2*b^10 - 7*a^8*b^2*c^2 - 14*a^4*b^6*c^2 + 5*a^2*b^8*c^2 - 4*b^10*c^2 + 4*a^8*c^4 + 33*a^4*b^4*c^4 + 16*b^8*c^4 - 2*a^6*c^6 - 14*a^4*b^2*c^6 - 32*b^6*c^6 - 4*a^4*c^8 + 5*a^2*b^2*c^8 + 16*b^4*c^8 + 2*a^2*c^10 - 4*b^2*c^10) : :

X(14684) = singular focus of the cubic K793.

X(14684) lies on the McCay circumcircle and these lines: {182, 12525}, {381, 9175}, {538, 1153}, {5970, 11842}

X(14685) =  SINGULAR FOCUS OF THE CUBIC K816

Barycentrics    a^2*(a^10 - 7*a^6*b^4 + 11*a^4*b^6 - 6*a^2*b^8 + b^10 + 11*a^6*b^2*c^2 - 10*a^4*b^4*c^2 - 7*a^2*b^6*c^2 + 6*b^8*c^2 - 7*a^6*c^4 - 10*a^4*b^2*c^4 + 26*a^2*b^4*c^4 - 7*b^6*c^4 + 11*a^4*c^6 - 7*a^2*b^2*c^6 - 7*b^4*c^6 - 6*a^2*c^8 + 6*b^2*c^8 + c^10) : :

Let L be a line tangent to the Moses-Parry circle. Let P be the isogonal conjugate of the trilinear pole of L. As L varies, P traces a parabola with focus X(14685) and directrix line X(3)X(541). (Randy Hutson, November 2, 2017)

X(14685) lies on the Brocard circle, the curves Q123 and K816, and on these lines: {2, 6795}, {3, 1495}, {6, 647}, {131, 12900}, {232, 8585}, {378, 841}, {381, 3258}, {842, 1995}, {1316, 1649}, {2132, 14385}, {2502, 8429}, {5467, 6090}, {5642, 8724}, {6322, 10163}, {9129, 9155}, {10745, 13202}

X(14685) = circumcircle-inverse of X(1495)
X(14685) = othocentroidal-circle inverse of X(3258)
X(14685) = Moses-radical-circle inverse of X(6)
X(14685) = second-Brocard-circle inverse of X(10568)
X(14685) = Parry-isodynamic-circle inverse of X(8429)
X(14685) = crossdifference of every pair of points on line {30, 9209}
X(14685) = Psi-transform of X(74)
X(14685) = X(6)-Hirst inverse of X(8675)
X(14685) = X(1495)-vertex conjugate of X(8675)

X(14686) =  SINGULAR FOCUS OF THE CUBIC K817

Barycentrics    a*(a^8 - a^7*b - a^6*b^2 + a^5*b^3 - a^4*b^4 + a^3*b^5 + a^2*b^6 - a*b^7 - a^7*c + a^6*b*c + a^5*b^2*c + 2*a^4*b^3*c - 5*a^3*b^4*c - a^2*b^5*c + 5*a*b^6*c - 2*b^7*c - a^6*c^2 + a^5*b*c^2 - a^2*b^4*c^2 + a*b^5*c^2 + a^5*c^3 + 2*a^4*b*c^3 + 6*a^2*b^3*c^3 - 5*a*b^4*c^3 + 2*b^5*c^3 - a^4*c^4 - 5*a^3*b*c^4 - a^2*b^2*c^4 - 5*a*b^3*c^4 + a^3*c^5 - a^2*b*c^5 + a*b^2*c^5 + 2*b^3*c^5 + a^2*c^6 + 5*a*b*c^6 - a*c^7 - 2*b*c^7) : :

X(14686) lies on the cubic K817 and these lines: {1, 6}, {123, 6667}, {378, 2691}, {1995, 2752}, {3309, 11472}

X(14687) =  SINGULAR FOCUS OF THE CUBIC K818

Barycentrics    a^2*(a^10 - 2*a^8*b^2 + 3*a^6*b^4 - 7*a^4*b^6 + 8*a^2*b^8 - 3*b^10 - 2*a^8*c^2 + 3*a^6*b^2*c^2 + 2*a^4*b^4*c^2 - 7*a^2*b^6*c^2 + 4*b^8*c^2 + 3*a^6*c^4 + 2*a^4*b^2*c^4 + 2*a^2*b^4*c^4 - b^6*c^4 - 7*a^4*c^6 - 7*a^2*b^2*c^6 - b^4*c^6 + 8*a^2*c^8 + 4*b^2*c^8 - 3*c^10) : :

X(14686) lies on the cubic K818 and these lines: {2, 11657}, {3, 6}, {110, 13558}, {122, 6723}, {378, 691}, {381, 2453}, {512, 11472}, {842, 1995}, {5191, 9970}, {5663, 9142}, {6795, 7422}, {8704, 9756}

X(14687) = circumcircle-inverse of X(32110)
X(14687) = {X(3),X(1351)}-harmonic conjugate of X(5467)

X(14688) =  SINGULAR FOCUS OF THE CUBIC K819

Barycentrics    a^2*(2*a^10 - 9*a^8*b^2 - 2*a^6*b^4 + 8*a^4*b^6 + b^10 - 9*a^8*c^2 + 50*a^6*b^2*c^2 - 15*a^4*b^4*c^2 - 10*b^8*c^2 - 2*a^6*c^4 - 15*a^4*b^2*c^4 - 48*a^2*b^4*c^4 + 25*b^6*c^4 + 8*a^4*c^6 + 25*b^4*c^6 - 10*b^2*c^8 + c^10) : :
X(14688) = X(111) - 3 X(5085) = X(11258) - 5 X(12017)

X(14688) lies on these lines: {3, 2854}, {6, 1296}, {111, 5085}, {126, 1503}, {1350, 10765}, {1428, 6019}, {2330, 3325}, {2780, 6593}, {2793, 5026}, {3589, 5512}, {11258, 12017}, {11422, 12149}

X(14686) = midpoint of X(i) and X(j) for these {i,j}: {6, 1296}, {1350, 10765}
X(14686) = reflection of X(5512) in X(3589)

X(14689) =  SINGULAR FOCUS OF THE CUBIC K824

Barycentrics    (a^2 - b^2 - c^2)*(4*a^12 - 4*a^10*b^2 - a^8*b^4 + 2*a^6*b^6 - 4*a^4*b^8 + 2*a^2*b^10 + b^12 - 4*a^10*c^2 + 6*a^8*b^2*c^2 - 2*a^6*b^4*c^2 + 6*a^4*b^6*c^2 - 2*a^2*b^8*c^2 - 4*b^10*c^2 - a^8*c^4 - 2*a^6*b^2*c^4 - 4*a^4*b^4*c^4 + 7*b^8*c^4 + 2*a^6*c^6 + 6*a^4*b^2*c^6 - 8*b^6*c^6 - 4*a^4*c^8 - 2*a^2*b^2*c^8 + 7*b^4*c^8 + 2*a^2*c^10 - 4*b^2*c^10 + c^12) : :
X(14689) = 3 X(376) - X(1297), 3 X(5731) - X(10705), 3 X(10304) - X(10718), 3 X(127) - 2 X(10749), 3 X(3) - X(10749), 3 X(112) - X(12384), 3 X(20) + X(12384), 3 X(376) + X(13200), 5 X(3522) - X(13219), 3 X(165) - X(13280), 3 X(3534) + X(13310)

X(14689) lies on these lines: {2, 10735}, {3, 114}, {4, 6720}, {20, 112}, {30, 132}, {165, 13280}, {339, 10991}, {376, 1297}, {516, 11722}, {1657, 12918}, {2781, 9967}, {2848, 3184}, {3070, 13923}, {3071, 13992}, {3522, 13219}, {3534, 9530}, {4299, 13311}, {4302, 13312}, {5204, 13297}, {5217, 13296}, {5731, 10705}, {6200, 13918}, {6361, 13099}, {6396, 13985}, {7386, 9157}, {10304, 10718}
X(14689) = complement X(10735)
X(14689) = midpoint of X(i) and X(j) for these {i,j}: {20, 112}, {1297, 13200}, {1657, 12918}, {6361, 13099}
X(14689) = reflection of X(i) in X(j) for these {i,j}: {4, 6720}, {127, 3}
X(14689) = complement X(10735)
X(14689) = {X(376),X(13200)}-harmonic conjugate of X(1297)

X(14690) =  SINGULAR FOCUS OF THE CUBIC K826

Barycentrics    a*(2*a^9 - a^8*b - 6*a^7*b^2 + 4*a^6*b^3 + 6*a^5*b^4 - 6*a^4*b^5 - 2*a^3*b^6 + 4*a^2*b^7 - b^9 - a^8*c + 2*a^7*b*c + 5*a^6*b^2*c - 10*a^5*b^3*c - 5*a^4*b^4*c + 14*a^3*b^5*c - a^2*b^6*c - 6*a*b^7*c + 2*b^8*c - 6*a^7*c^2 + 5*a^6*b*c^2 + 11*a^4*b^3*c^2 - 2*a^3*b^4*c^2 - 17*a^2*b^5*c^2 + 8*a*b^6*c^2 + b^7*c^2 + 4*a^6*c^3 - 10*a^5*b*c^3 + 11*a^4*b^2*c^3 - 20*a^3*b^3*c^3 + 14*a^2*b^4*c^3 + 6*a*b^5*c^3 - 5*b^6*c^3 + 6*a^5*c^4 - 5*a^4*b*c^4 - 2*a^3*b^2*c^4 + 14*a^2*b^3*c^4 - 16*a*b^4*c^4 + 3*b^5*c^4 - 6*a^4*c^5 + 14*a^3*b*c^5 - 17*a^2*b^2*c^5 + 6*a*b^3*c^5 + 3*b^4*c^5 - 2*a^3*c^6 - a^2*b*c^6 + 8*a*b^2*c^6 - 5*b^3*c^6 + 4*a^2*c^7 - 6*a*b*c^7 + b^2*c^7 + 2*b*c^8 - c^9) : :
X(14690) = X(102) - 3 X(165) = X(151) + 3 X(9778) = 2 X(6711) - 3 X(10164) = 3 X(3576) - X(10703) = 3 X(5587) - X(10732) = 3 X(10165) - 2 X(11734) = 3 X(5657) - X(13532)

X(14690) lies on these lines: {3, 214}, {40, 109}, {55, 12016}, {102, 165}, {117, 516}, {124, 6684}, {151, 9778}, {484, 1845}, {517, 11700}, {946, 6718}, {1155, 1361}, {1633, 2950}, {1795, 5119}, {2818, 3579}, {2835, 3359}, {3040, 4640}, {3576, 10703}, {4301, 11727}, {5587, 10732}, {5657, 13532}, {6711, 10164}, {7991, 10696}, {10165, 11734}, {13528, 13539}

X(14690) = midpoint of X(i) and X(j) for these {i,j}: {40, 109}, {7991, 10696}
X(14690) = reflection of X(i) in X(j) for these {i,j}: {124, 6684}, {946, 6718}, {4301, 11727}, {11713, 3}
X(14690) = circumcircle-inverse of X(12332)
X(14690) = X(3738)-vertex conjugate of X(12332)

X(14691) =  SINGULAR FOCUS OF THE CUBIC K828

Barycentrics    a^2*(a^8*b^4 + a^6*b^6 - 2*a^8*b^2*c^2 - a^6*b^4*c^2 - a^4*b^6*c^2 + a^2*b^8*c^2 + a^8*c^4 - a^6*b^2*c^4 - a^4*b^4*c^4 + 2*a^2*b^6*c^4 - b^8*c^4 + a^6*c^6 - a^4*b^2*c^6 + 2*a^2*b^4*c^6 - b^6*c^6 + a^2*b^2*c^8 - b^4*c^8) : :

X(14691) lies on these lines: {32, 99}, {729, 3098}, {2076, 9431}, {3094, 9427}, {3972, 6374}, {5027, 9998}, {5106, 9999}

X(14692) =  SINGULAR FOCUS OF THE CUBIC K837

Barycentrics    a^8 + 4*a^6*b^2 - 5*a^4*b^4 + a^2*b^6 - b^8 + 4*a^6*c^2 - 9*a^4*b^2*c^2 + 4*a^2*b^4*c^2 - 2*b^6*c^2 - 5*a^4*c^4 + 4*a^2*b^2*c^4 + 6*b^4*c^4 + a^2*c^6 - 2*b^2*c^6 - c^8 : :
X(14692) = 3 X(4) - 5 X(147) = 7 X(4) - 5 X(148) = 7 X(147) - 3 X(148) = 5 X(99) - 4 X(548) = 5 X(98) - 6 X(549) = 10 X(114) - 9 X(5055) = 10 X(115) - 11 X(5072) = 7 X(114) - 6 X(5461) = 4 X(148) - 7 X(6033) = 4 X(4) - 5 X(6033) = 4 X(147) - 3 X(6033) = 4 X(5066) - 5 X(6054) = 6 X(148) - 7 X(6321) = 6 X(4) - 5 X(6321) = 3 X(6033) - 2 X(6321) = 4 X(549) - 5 X(8724) = 2 X(98) - 3 X(8724) = 5 X(5984) - 9 X(10304) = 8 X(5461) - 7 X(11632) = 6 X(5055) - 5 X(11632) = 4 X(114) - 3 X(11632) = 4 X(12007) - 5 X(12177) = 7 X(3526) - 5 X(12188) = 4 X(6036) - 3 X(12188) = 3 X(3534) - 5 X(13188)

X(14692) lies on these lines: {4, 147}, {98, 549}, {99, 548}, {114, 5055}, {115, 5072}, {542, 1350}, {1569, 11646}, {1657, 7855}, {3526, 6036}, {3627, 7905}, {3628, 7859}, {4027, 14036}, {5066, 6054}, {5984, 10304}, {7936, 11257}, {12007, 12177}

X(14692) = reflection of X(6321) in X(147) for these {i,j}: {6321, 147}
X(14692) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (147, 6321, 6033)

X(14693) =  SINGULAR FOCUS OF THE CUBIC K869

Barycentrics    2*a^8 - 6*a^6*b^2 + 6*a^4*b^4 - 3*a^2*b^6 + b^8 - 6*a^6*c^2 + 2*a^2*b^4*c^2 - 3*b^6*c^2 + 6*a^4*c^4 + 2*a^2*b^2*c^4 + 4*b^4*c^4 - 3*a^2*c^6 - 3*b^2*c^6 + c^8 : :
X(14793) = X(316) - 5 X(1656) = X(1352) + 3 X(1691) = X(1353) - 3 X(1692) = 3 X(2) + X(2080) = X(549) - 3 X(5215) = X(5184) + 3 X(5886) = X(8724) + 3 X(8859) = 7 X(3526) + X(9301) = 3 X(5) - X(13449) = 3 X(187) + X(13449) = X(6321) + 3 X(13586) = X(5104) + 3 X(14561) = X(13188) + 3 X(14568)

X(14793) lies on these lines:
{2, 2080}, {3, 7790}, {5, 187}, {30, 5461}, {140, 143}, {230, 2021}, {316, 1656}, {542, 8590}, {547, 3849}, {549, 5215}, {625, 3628}, {632, 7889}, {754, 6721}, {1352, 1691}, {1353, 1692}, {1506, 10631}, {1513, 12042}, {2030, 3564}, {2459, 3070}, {2460, 3071}, {3095, 7907}, {3526, 7846}, {5031, 7815}, {5104, 14561}, {5184, 5886}, {6321, 13586}, {7684, 13349}, {7685, 13350}, {7806, 11171}, {8724, 8859}, {13188, 14568}

X(14693) = midpoint of X(i) and X(j) for these {i,j}: {5, 187}, {1513, 12042}, {7684, 13349}, {7685, 13350}
X(14693) = reflection of X(625) in X(3628)

X(14694) =  SINGULAR FOCUS OF THE CUBIC K870

Barycentrics    4*a^10 - 8*a^8*b^2 + 4*a^6*b^4 + 7*a^4*b^6 - 8*a^2*b^8 + b^10 - 8*a^8*c^2 + 16*a^6*b^2*c^2 - 15*a^4*b^4*c^2 + 22*a^2*b^6*c^2 - 5*b^8*c^2 + 4*a^6*c^4 - 15*a^4*b^2*c^4 - 24*a^2*b^4*c^4 + 4*b^6*c^4 + 7*a^4*c^6 + 22*a^2*b^2*c^6 + 4*b^4*c^6 - 8*a^2*c^8 - 5*b^2*c^8 + c^10 : :

X(14694) lies on the cubic K870 and these lines: {2, 3}, {111, 11632}, {351, 2793}, {511, 1641}, {542, 1648}, {1640, 11656}, {6033, 9759}, {8724, 9775}, {9169, 11179}
X(14694) = midpoint of X(2) and X(7417)
X(14694) = Hutson-Parry-circle (= Yiu-circle-inverse) of X(9172)
X(14694) = crossdifference of every pair of points on line {647, 5653}
X(14694) = intersection of tangents to Hutson-Parry circle at X(2) and X(111)
X(14694) = intersection of tangents to circle {{X(2),X(110),X(2770),X(5463),X(5464)}} at X(2) and X(2770)
X(14694) = pole of line X(2)X(99) wrt Hutson-Parry circle
X(14694) = pole of line X(2)X(691) wrt circle {{X(2),X(110),X(2770),X(5463),X(5464)}}

X(14695) =  SINGULAR FOCUS OF THE CUBIC K878

Barycentrics    (b - c)*(b + c)*(-3*a^14 + 11*a^12*b^2 - 15*a^10*b^4 + 11*a^8*b^6 - 9*a^6*b^8 + 9*a^4*b^10 - 5*a^2*b^12 + b^14 + 11*a^12*c^2 - 28*a^10*b^2*c^2 + 25*a^8*b^4*c^2 - 6*a^6*b^6*c^2 - 9*a^4*b^8*c^2 + 10*a^2*b^10*c^2 - 3*b^12*c^2 - 15*a^10*c^4 + 25*a^8*b^2*c^4 - 13*a^6*b^4*c^4 + 5*a^4*b^6*c^4 - 9*a^2*b^8*c^4 + 3*b^10*c^4 + 11*a^8*c^6 - 6*a^6*b^2*c^6 + 5*a^4*b^4*c^6 + 8*a^2*b^6*c^6 - b^8*c^6 - 9*a^6*c^8 - 9*a^4*b^2*c^8 - 9*a^2*b^4*c^8 - b^6*c^8 + 9*a^4*c^10 + 10*a^2*b^2*c^10 + 3*b^4*c^10 - 5*a^2*c^12 - 3*b^2*c^12 + c^14) : :
X(14695) = X(265) - 3 X(13291) = 3 X(10190) - 4 X(13392)

X(14695) lies on these lines: {110, 8151}, {146, 1499}, {265, 13291}, {525, 2931}, {526, 12236}, {542, 12076}, {690, 6036}, {3448, 10279}, {10190, 13392}

X(14695) = reflection of X(i) in X(j) for these {i,j}: {3448, 10279}, {8151, 110}

X(14696) =  SINGULAR FOCUS OF THE CUBIC K882

Barycentrics    a^2*(b - c)*(b + c)*(a^8 - 5*a^4*b^4 + 6*a^2*b^6 - 2*b^8 - a^4*b^2*c^2 + b^6*c^2 - 5*a^4*c^4 + 6*a^2*c^6 + b^2*c^6 - 2*c^8) : :

X(14696) lies on these lines: {2, 690}, {110, 14590}, {351, 924}, {6368, 9979}

X(14697) =  SINGULAR FOCUS OF THE CUBIC K884

Barycentrics    (b - c)*(b + c)*(-4*a^10 + 7*a^8*b^2 - a^6*b^4 - 2*a^4*b^6 - a^2*b^8 + b^10 + 7*a^8*c^2 - 16*a^6*b^2*c^2 + 7*a^4*b^4*c^2 + 4*a^2*b^6*c^2 - 2*b^8*c^2 - a^6*c^4 + 7*a^4*b^2*c^4 - 8*a^2*b^4*c^4 + b^6*c^4 - 2*a^4*c^6 + 4*a^2*b^2*c^6 + b^4*c^6 - a^2*c^8 - 2*b^2*c^8 + c^10) : :
X(14697) = X(9138) - 3 X(9185) = 3 X(351) - X(13290) = X(13290) + 3 X(13291)

X(14697) lies on these lines: {2, 690}, {107, 110}, {113, 1560}, {351, 13290}, {468, 525}, {542, 1637}, {1640, 11656}, {2799, 5642}, {5972, 14417}, {6333, 7664}, {13114, 14345}

X(14697) = midpoint of X(i) and X(j) for these {i,j}: {110, 9979}, {351, 13291}
X(14697) = reflection of X(14417) in X(5972)
X(14697) = crossdifference of every pair of points on line {2502, 3269}
X(14697) = X(16163)-of-1st-Parry-triangle
X(14697) = X(125)-of-2nd-Parry-triangle

X(14698) =  SINGULAR FOCUS OF THE CUBIC K885

Barycentrics    (b - c)*(b + c)*(-3*a^10 + 7*a^8*b^2 - 6*a^6*b^4 + 4*a^4*b^6 - 3*a^2*b^8 + b^10 + 7*a^8*c^2 - 12*a^6*b^2*c^2 + 5*a^4*b^4*c^2 + 4*a^2*b^6*c^2 - 2*b^8*c^2 - 6*a^6*c^4 + 5*a^4*b^2*c^4 - 7*a^2*b^4*c^4 + b^6*c^4 + 4*a^4*c^6 + 4*a^2*b^2*c^6 + b^4*c^6 - 3*a^2*c^8 - 2*b^2*c^8 + c^10) : :
X(14698) = 2 X(9138) - 3 X(9185) = 3 X(9123) - 2 X(13290)

X(14698) lies on these lines: {2, 690}, {110, 925}, {246, 3134}, {526, 9979}, {858, 3566}, {3448, 9134}, {9123, 13290}

X(14698) = reflection of X(i) in X(j) for these {i,j}: {3448, 9134}, {9131, 110}, {9979, 13291}

X(14699) =  SINGULAR FOCUS OF THE CUBIC K887

Barycentrics    a^2*(b - c)*(b + c)*(4*a^14 - 2*a^10*b^4 - 8*a^8*b^6 - 8*a^6*b^8 + 16*a^4*b^10 + 6*a^2*b^12 - 8*b^14 - 5*a^10*b^2*c^2 - 8*a^8*b^4*c^2 + 33*a^6*b^6*c^2 - 11*a^4*b^8*c^2 - 29*a^2*b^10*c^2 + 18*b^12*c^2 - 2*a^10*c^4 - 8*a^8*b^2*c^4 + 28*a^6*b^4*c^4 - 23*a^4*b^6*c^4 + 10*a^2*b^8*c^4 - 5*b^10*c^4 - 8*a^8*c^6 + 33*a^6*b^2*c^6 - 23*a^4*b^4*c^6 + 9*a^2*b^6*c^6 + b^8*c^6 - 8*a^6*c^8 - 11*a^4*b^2*c^8 + 10*a^2*b^4*c^8 + b^6*c^8 + 16*a^4*c^10 - 29*a^2*b^2*c^10 - 5*b^4*c^10 + 6*a^2*c^12 + 18*b^2*c^12 - 8*c^14) : :

X(14699) lies on the Yiu circle and these lines: {3, 11621}, {691, 1576}

X(14700) =  SINGULAR FOCUS OF THE CUBIC K895

Barycentrics    a^2*(a^4*b^4 + a^2*b^6 - a^4*b^2*c^2 - 2*a^2*b^4*c^2 - 2*b^6*c^2 + a^4*c^4 - 2*a^2*b^2*c^4 + 5*b^4*c^4 + a^2*c^6 - 2*b^2*c^6) : :

X(14700) is the intersection, other than X(4), of the orthocentroidal circle and circle {X(4),X(194),X(3557),X(3558)} (Randy Hutson, November 2, 2017)

X(14700) lies on these lines: {2, 39}, {4, 2489}, {6, 6787}, {32, 691}, {61, 14174}, {62, 14180}, {575, 6235}, {671, 1084}, {2142, 7760}, {3053, 11634}, {6034, 9144}, {6792, 11182}, {7752, 9152}, {7772, 11638}, {9178, 14263}, {9605, 11637}

X(14700) = polar-circle inverse of X(2489)
X(14700) = orthoptic-circle-of-Steiner-inellipe-inverse of X(3291)
X(14700) = Psi-transform of X(3124)

X(14701) =  SINGULAR FOCUS OF THE CUBIC K897

Barycentrics    a^2*(a^10 - a^6*b^4 - a^4*b^6 - 2*a^2*b^8 - b^10 - 4*a^6*b^2*c^2 + 3*a^4*b^4*c^2 + 2*a^2*b^6*c^2 - a^6*c^4 + 3*a^4*b^2*c^4 + 5*a^2*b^4*c^4 - b^6*c^4 - a^4*c^6 + 2*a^2*b^2*c^6 - b^4*c^6 - 2*a^2*c^8 - c^10) : :

X(14701) lies on these lines: {2, 4048}, {6, 11641}, {755, 7953}

X(14701) = Psi-transform of X(1627)

X(14702) =  SINGULAR FOCUS OF THE CUBIC K903

Barycentrics    a^2*(a^14 - 14*a^10*b^4 + 35*a^8*b^6 - 35*a^6*b^8 + 14*a^4*b^10 - b^14 + 15*a^10*b^2*c^2 - 34*a^8*b^4*c^2 + 11*a^6*b^6*c^2 + 21*a^4*b^8*c^2 - 14*a^2*b^10*c^2 + b^12*c^2 - 14*a^10*c^4 - 34*a^8*b^2*c^4 + 78*a^6*b^4*c^4 - 45*a^4*b^6*c^4 + 12*a^2*b^8*c^4 + 3*b^10*c^4 + 35*a^8*c^6 + 11*a^6*b^2*c^6 - 45*a^4*b^4*c^6 + 4*a^2*b^6*c^6 - 3*b^8*c^6 - 35*a^6*c^8 + 21*a^4*b^2*c^8 + 12*a^2*b^4*c^8 - 3*b^6*c^8 + 14*a^4*c^10 - 14*a^2*b^2*c^10 + 3*b^4*c^10 + b^2*c^12 - c^14) : :

X(14702) lies on the cubic K903 and these lines: {3, 541}, {23, 477}, {9003, 12584}

X(14702) = circumcircle-inverse of X(5655)
X(14702) = X(5655)-vertex conjugate of X(9003)

X(14703) =  SINGULAR FOCUS OF THE CUBIC K904

Barycentrics    a^2*(a^14 - 2*a^12*b^2 - 4*a^10*b^4 + 15*a^8*b^6 - 15*a^6*b^8 + 4*a^4*b^10 + 2*a^2*b^12 - b^14 - 2*a^12*c^2 + 9*a^10*b^2*c^2 - 12*a^8*b^4*c^2 + 12*a^4*b^8*c^2 - 9*a^2*b^10*c^2 + 2*b^12*c^2 - 4*a^10*c^4 - 12*a^8*b^2*c^4 + 26*a^6*b^4*c^4 - 16*a^4*b^6*c^4 + 6*a^2*b^8*c^4 + 15*a^8*c^6 - 16*a^4*b^4*c^6 + 2*a^2*b^6*c^6 - b^8*c^6 - 15*a^6*c^8 + 12*a^4*b^2*c^8 + 6*a^2*b^4*c^8 - b^6*c^8 + 4*a^4*c^10 - 9*a^2*b^2*c^10 + 2*a^2*c^12 + 2*b^2*c^12 - c^14) : :

X(14703) lies on the tangential circle, the cubic K904, and these lines: {3, 113}, {22, 1294}, {24, 107}, {25, 133}, {157, 1605}, {186, 5667}, {187, 9412}, {378, 10152}, {399, 5502}, {577, 3165}, {1511, 1576}, {1609, 2079}, {2931, 9033}, {6642, 6716}, {7731, 14385}, {9530, 14070}

X(14703) = circumcircle-inverse of X(113)
X(14703) = Stammler-circle-inverse of X(38790)
X(14703) = X(113)-vertex conjugate of X(9033)
X(14703) = X(106)-of-tangential-triangle if ABC is acute

X(14704) =  SINGULAR FOCUS OF THE CUBIC K261a

Barycentrics    a^2(Sqrt(3) (a^2-b^2-c^2) (a^4-a^2 b^2+b^4-a^2 c^2-b^2 c^2+c^4)+2 (a^4-a^2 b^2-3 b^4-a^2 c^2+7 b^2 c^2-3 c^4) S) : :

X(14704) lies on the Parry circle, the curve Q041, and these lines: {2,5469}, {3,8450}, {16,3124}, {61,110}, {111,10645}, {9147,9201}

X(14704) = Psi-transform of X(62)

X(14705) =  SINGULAR FOCUS OF THE CUBIC K261b

Barycentrics    a^2(Sqrt(3) (a^2-b^2-c^2) (a^4-a^2 b^2+b^4-a^2 c^2-b^2 c^2+c^4)-2 (a^4-a^2 b^2-3 b^4-a^2 c^2+7 b^2 c^2-3 c^4) S) : :

X(14705) lies on the Parry circle, the curve Q041, and these lines: {2,5470}, {3,8450}, {15,3124}, {62,110}, {111,10646}, {9147,9200}

X(14705) = Psi-transform of X(61)

X(14706) =  SINGULAR FOCUS OF THE CUBIC K067

Barycentrics    a^2 (a^16-7 a^14 b^2+19 a^12 b^4-23 a^10 b^6+5 a^8 b^8+19 a^6 b^10-23 a^4 b^12+11 a^2 b^14-2 b^16-7 a^14 c^2+37 a^12 b^2 c^2-75 a^10 b^4 c^2+83 a^8 b^6 c^2-79 a^6 b^8 c^2+81 a^4 b^10 c^2-55 a^2 b^12 c^2+15 b^14 c^2+19 a^12 c^4-75 a^10 b^2 c^4+87 a^8 b^4 c^4-27 a^6 b^6 c^4-45 a^4 b^8 c^4+87 a^2 b^10 c^4-46 b^12 c^4-23 a^10 c^6+83 a^8 b^2 c^6-27 a^6 b^4 c^6+19 a^4 b^6 c^6-43 a^2 b^8 c^6+81 b^10 c^6+5 a^8 c^8-79 a^6 b^2 c^8-45 a^4 b^4 c^8-43 a^2 b^6 c^8-96 b^8 c^8+19 a^6 c^10+81 a^4 b^2 c^10+87 a^2 b^4 c^10+81 b^6 c^10-23 a^4 c^12-55 a^2 b^2 c^12-46 b^4 c^12+11 a^2 c^14+15 b^2 c^14-2 c^16 : :

X(14706) lies the curve Q041, and these lines: {3,14668}, {54,5898}, {140,930}

X(14706) = circumcircle-inverse of X(14668)
X(14706) = Psi-transform of X(195)

X(14707) =  16th HATZIPOLAKIS-MOSES-EULER POINT

Barycentrics    3 a^36 b^4-39 a^34 b^6+228 a^32 b^8-780 a^30 b^10+1680 a^28 b^12-2184 a^26 b^14+1092 a^24 b^16+1716 a^22 b^18-4290 a^20 b^20+4290 a^18 b^22-1716 a^16 b^24-1092 a^14 b^26+2184 a^12 b^28-1680 a^10 b^30+780 a^8 b^32-228 a^6 b^34+39 a^4 b^36-3 a^2 b^38+10 a^36 b^2 c^2-145 a^34 b^4 c^2+882 a^32 b^6 c^2-2986 a^30 b^8 c^2+6141 a^28 b^10 c^2-7622 a^26 b^12 c^2+4965 a^24 b^14 c^2-866 a^22 b^16 c^2+869 a^20 b^18 c^2-4312 a^18 b^20 c^2+3949 a^16 b^22 c^2+2850 a^14 b^24 c^2-9689 a^12 b^26 c^2+10470 a^10 b^28 c^2-6529 a^8 b^30 c^2+2538 a^6 b^32 c^2-595 a^4 b^34 c^2+73 a^2 b^36 c^2-3 b^38 c^2+3 a^36 c^4-145 a^34 b^2 c^4+1316 a^32 b^4 c^4-5418 a^30 b^6 c^4+12216 a^28 b^8 c^4-15408 a^26 b^10 c^4+8907 a^24 b^12 c^4+1975 a^22 b^14 c^4-7463 a^20 b^16 c^4+8577 a^18 b^18 c^4-9164 a^16 b^20 c^4+3100 a^14 b^22 c^4+12006 a^12 b^24 c^4-23586 a^10 b^26 c^4+21507 a^8 b^28 c^4-11481 a^6 b^30 c^4+3638 a^4 b^32 c^4-622 a^2 b^34 c^4+42 b^36 c^4-39 a^34 c^6+882 a^32 b^2 c^6-5418 a^30 b^4 c^6+15158 a^28 b^6 c^6-21282 a^26 b^8 c^6+12037 a^24 b^10 c^6+4905 a^22 b^12 c^6-11061 a^20 b^14 c^6+4875 a^18 b^16 c^6+4971 a^16 b^18 c^6-11454 a^14 b^20 c^6+3138 a^12 b^22 c^6+19666 a^10 b^24 c^6-34263 a^8 b^26 c^6+27431 a^6 b^28 c^6-12099 a^4 b^30 c^6+2820 a^2 b^32 c^6-267 b^34 c^6+228 a^32 c^8-2986 a^30 b^2 c^8+12216 a^28 b^4 c^8-21282 a^26 b^6 c^8+13390 a^24 b^8 c^8+5662 a^22 b^10 c^8-10450 a^20 b^12 c^8+472 a^18 b^14 c^8+5425 a^16 b^16 c^8+949 a^14 b^18 c^8-10747 a^12 b^20 c^8+2473 a^10 b^22 c^8+23737 a^8 b^24 c^8-36453 a^6 b^26 c^8+24201 a^4 b^28 c^8-7843 a^2 b^30 c^8+1008 b^32 c^8-780 a^30 c^10+6141 a^28 b^2 c^10-15408 a^26 b^4 c^10+12037 a^24 b^6 c^10+5662 a^22 b^8 c^10-10298 a^20 b^10 c^10+266 a^18 b^12 c^10+2380 a^16 b^14 c^10+2179 a^14 b^16 c^10+1103 a^12 b^18 c^10-10625 a^10 b^20 c^10+1339 a^8 b^22 c^10+23863 a^6 b^24 c^10-29442 a^4 b^26 c^10+14043 a^2 b^28 c^10-2460 b^30 c^10+1680 a^28 c^12-7622 a^26 b^2 c^12+8907 a^24 b^4 c^12+4905 a^22 b^6 c^12-10450 a^20 b^8 c^12+266 a^18 b^10 c^12+2758 a^16 b^12 c^12-996 a^14 b^14 c^12+2557 a^12 b^16 c^12+1337 a^10 b^18 c^12-10255 a^8 b^20 c^12-843 a^6 b^22 c^12+19803 a^4 b^24 c^12-15911 a^2 b^26 c^12+3864 b^28 c^12-2184 a^26 c^14+4965 a^24 b^2 c^14+1975 a^22 b^4 c^14-11061 a^20 b^6 c^14+472 a^18 b^8 c^14+2380 a^16 b^10 c^14-996 a^14 b^12 c^14-1104 a^12 b^14 c^14+1945 a^10 b^16 c^14+1501 a^8 b^18 c^14-9339 a^6 b^20 c^14-3821 a^4 b^22 c^14+10047 a^2 b^24 c^14-3276 b^26 c^14+1092 a^24 c^16-866 a^22 b^2 c^16-7463 a^20 b^4 c^16+4875 a^18 b^6 c^16+5425 a^16 b^8 c^16+2179 a^14 b^10 c^16+2557 a^12 b^12 c^16+1945 a^10 b^14 c^16+4366 a^8 b^16 c^16+4512 a^6 b^18 c^16-4673 a^4 b^20 c^16-1757 a^2 b^22 c^16-624 b^24 c^16+1716 a^22 c^18+869 a^20 b^2 c^18+8577 a^18 b^4 c^18+4971 a^16 b^6 c^18+949 a^14 b^8 c^18+1103 a^12 b^10 c^18+1337 a^10 b^12 c^18+1501 a^8 b^14 c^18+4512 a^6 b^16 c^18+5898 a^4 b^18 c^18-847 a^2 b^20 c^18+6006 b^22 c^18-4290 a^20 c^20-4312 a^18 b^2 c^20-9164 a^16 b^4 c^20-11454 a^14 b^6 c^20-10747 a^12 b^8 c^20-10625 a^10 b^10 c^20-10255 a^8 b^12 c^20-9339 a^6 b^14 c^20-4673 a^4 b^16 c^20-847 a^2 b^18 c^20-8580 b^20 c^20+4290 a^18 c^22+3949 a^16 b^2 c^22+3100 a^14 b^4 c^22+3138 a^12 b^6 c^22+2473 a^10 b^8 c^22+1339 a^8 b^10 c^22-843 a^6 b^12 c^22-3821 a^4 b^14 c^22-1757 a^2 b^16 c^22+6006 b^18 c^22-1716 a^16 c^24+2850 a^14 b^2 c^24+12006 a^12 b^4 c^24+19666 a^10 b^6 c^24+23737 a^8 b^8 c^24+23863 a^6 b^10 c^24+19803 a^4 b^12 c^24+10047 a^2 b^14 c^24-624 b^16 c^24-1092 a^14 c^26-9689 a^12 b^2 c^26-23586 a^10 b^4 c^26-34263 a^8 b^6 c^26-36453 a^6 b^8 c^26-29442 a^4 b^10 c^26-15911 a^2 b^12 c^26-3276 b^14 c^26+2184 a^12 c^28+10470 a^10 b^2 c^28+21507 a^8 b^4 c^28+27431 a^6 b^6 c^28+24201 a^4 b^8 c^28+14043 a^2 b^10 c^28+3864 b^12 c^28-1680 a^10 c^30-6529 a^8 b^2 c^30-11481 a^6 b^4 c^30-12099 a^4 b^6 c^30-7843 a^2 b^8 c^30-2460 b^10 c^30+780 a^8 c^32+2538 a^6 b^2 c^32+3638 a^4 b^4 c^32+2820 a^2 b^6 c^32+1008 b^8 c^32-228 a^6 c^34-595 a^4 b^2 c^34-622 a^2 b^4 c^34-267 b^6 c^34+39 a^4 c^36+73 a^2 b^2 c^36+42 b^4 c^36-3 a^2 c^38-3 b^2 c^38 : :

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 26652

X(14707) lies this line: {2,3}

X(14708) =  MIDPOINT OF X(113) AND X(185)

Barycentrics    a^2 (-a^12 (b^2+c^2)+4 a^10 (b^4+c^4)+a^8 (-5 b^6+2 b^4 c^2+2 b^2 c^4-5 c^6)+(b^2-c^2)^4 (b^6+2 b^4 c^2+2 b^2 c^4+c^6)+2 a^6 (3 b^6 c^2-8 b^4 c^4+3 b^2 c^6)-2 a^2 (b^2-c^2)^2 (2 b^8-b^6 c^2-3 b^4 c^4-b^2 c^6+2 c^8)+a^4 (5 b^10-15 b^8 c^2+12 b^6 c^4+12 b^4 c^6-15 b^2 c^8+5 c^10)) : :
X(14708) = 3*X(2)+X(7722) = X(4)-3*X(16222) = 3*X(51)-X(12295) = X(74)-5*X(10574) = X(110)+3*X(5890) = 3*X(110)+X(12284) = X(113)-3*X(16223) = X(125)-3*X(9730) = 2*X(140)+X(13148) = X(185)+3*X(16223) = X(265)-5*X(37481) = 3*X(381)-X(12292) = X(974)+2*X(11561) = 9*X(5890)-X(12284) = 3*X(9730)+X(11562) = 4*X(12006)-X(15738) = X(12162)-3*X(36518) = 3*X(13363)-2*X(15088) = 2*X(13630)+X(25711) = X(21650)-3*X(23515)

See Antreas Hatzipolakis and Angel Montesdeoca, Hyacinthos 26654

X(14708) lies on these lines: {2,7722}, {3,1986}, {4,16222}, {5,113}, {6,12302}, {24,20773}, {30,1112}, {51,12295}, {52,16163}, {74,7526}, {110,5890}, {140,12358}, {156,20771}, {182,2781}, {265,18912}, {381,12292}, {389,11800}, {399,6642}, {511,38726}, {541,17855}, {542,11806}, {546,12133}, {549,13416}, {568,12121}, {569,32607}, {578,12901}, {631,12219}, {1147,1511}, {1154,15646}, {1353,14984}, {1539,13491}, {1656,22584}, {2777,11557}, {2807,11723}, {2931,9786}, {2979,15036}, {3024,37729}, {3043,22467}, {3047,15032}, {3448,18420}, {3567,10733}, {5020,12308}, {5095,37511}, {5462,7687}, {5504,12161}, {5562,38793}, {5889,15035}, {5892,6723}, {5907,12900}, {5946,7706}, {5972,13754}, {6000,23323}, {6243,38723}, {6689,6699}, {7401,12317}, {7727,37696}, {7728,12824}, {7731,15055}, {8703,9967}, {9140,15102}, {9818,10620}, {10111,31833}, {10272,16238}, {10575,13202}, {10721,15072}, {11412,15051}, {11436,19469}, {11702,22962}, {11720,31728}, {11801,12099}, {11819,15473}, {12084,15472}, {12140,31830}, {12233,23306}, {12270,14644}, {12273,15034}, {12281,15045}, {12606,38448}, {12825,14643}, {12888,19366}, {13201,20791}, {13382,16534}, {13417,16111}, {14448,38727}, {15012,36253}, {15462,19139}, {16105,34584}, {17835,37514}, {17847,37475}, {18388,33547}, {18436,38794}, {18580,38728}, {19457,36752}, {19470,37697}, {21649,30714}

X(14708) = midpoint of X(i) and X(j) for these {i,j}: {3, 1986}, {52, 16163}, {113, 185}, {125, 11562}, {974, 25711}, {1511, 6102}, {1539, 13491}, {5095, 37511}, {7722, 7723}, {7728, 17854}, {10575, 13202}, {11561, 13630}, {11720, 31728}, {12358, 13148}, {12825, 34783}, {13417, 16111}, {21649, 30714}
X(14708) = reflection of X(i) in X(j) for these (i,j): (5, 9826), (974, 13630), (5907, 12900), (6699, 9729), (7687, 5462), (10113, 11746), (10264, 16270), (11819, 15473), (12133, 546), (12140, 31830), (12236, 389), (12358, 140), (15738, 20304), (20304, 12006), (25711, 11561)
X(14708) = complement of X(7723)
X(14708) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 7722, 7723), (3, 15463, 25487), (185, 16223, 113), (5946, 10113, 11746), (9730, 11562, 125), (12270, 15043, 14644), (12281, 15045, 15059), (12824, 17854, 7728), (14643, 34783, 12825)

X(14709) =  ISOGONAL CONJUGATE OF X(14374)

Barycentrics    a^2 (a^2+b^2-c^2) (a^2-b^2+c^2) ((-a^2+b^2+c^2)^2+b^2 c^2 (1-J)) : :

Let U = X(14709) and V = X(14710). Peter Moses noted (October 9,, 2017) the following relations on the Euler line:

X(5) = {U,V}-harmonic conjugate of X(3)
X(3520) = {U,V}-harmonic conjugate of X(4)
X(7527) = {U,V}-harmonic conjugate of X(20)
X(6143) = {U,V}-harmonic conjugate of X(186)
X(10226) = {U,V}-harmonic conjugate of X(381)
X(14130) = midpoint of U and V.

Let H=X(4). Let Ha be the reflection of H in the Euler line of triangle BCH. Define Hb and Hc cyclically. Triangle HaHbHc is perspective to ABC at X(252). X(14709) is the insimilicenter of the circumcircles of ABC and HaHbHc. The exsimilicenter is X(14710). (Randy Hutson, November 2, 2017)

X(14709) lies on these lines: {2,3}, {54,2574}, {74,14375}, {185,13414}, {572,2577}, {580,1823}, {1092,8115}, {5889,8116}, {10288,14385}, {13434,14374}

X(14709) = reflection of X(14710) in X(14130)
X(14709) = isogonal conjugate of X(14374)
X(14709) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 4, 1114), (3, 1344, 4), (186, 1313, 1114), (378, 1344, 1346), (1312, 1885, 4), (1344, 3516, 3091)

X(14710) =  ISOGONAL CONJUGATE OF X(14375)

Barycentrics    a^2 (a^2+b^2-c^2) (a^2-b^2+c^2) ((-a^2+b^2+c^2)^2+b^2 c^2 (1+J)) : :

See X(14709)

Let HaHbHc be as at X(14709). Then X(14710) is the exsimilicenter of the circumcircles of ABC and HaHbHc. (Randy Hutson, November 2, 2017)

X(14710) lies on these lines: {2,3}, {54,2575}, {74,14374}, {185,13415}, {572,2576}, {580,1822}, {1092,8116}, {5889,8115}, {10287,14385}, {13434,14375}

X(14710) = reflection of X(14709) in X(14130)
X(14710) = isogonal conjugate of X(14375)
X(14710) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 4, 1113), (3, 1345, 4), (186, 1312, 1113), (378, 1345, 1347), (1313, 1885, 4), (1345, 3516, 3091)

X(14711) =  COMPLEMENT OF X(11055)

Barycentrics    a^2 b^2+a^2 c^2-8 b^2 c^2 : :
X(14711) = X(2) - X(39) + X(76)
X(14711) = 4 X(2) - 3 X(39) = X(39) - 4 X(76) = X(2) - 3 X(76) = 7 X(39) - 4 X(194) = 7 X(2) - 3 X(194) = 7 X(76) - X(194) = 5 X(194) - 14 X(3934) = 5 X(39) - 8 X(3934) = 5 X(2) - 6 X(3934) = 5 X(76) - 2 X(3934) = 2 X(3534) - 3 X(5188) = 2 X(3845) - 3 X(6248) = 13 X(39) - 16 X(6683) = 13 X(2) - 12 X(6683) = 13 X(3934) - 10 X(6683) = 13 X(76) - 4 X(6683) = 5 X(194) - 7 X(7757) = 5 X(39) - 4 X(7757) = 5 X(2) - 3 X(7757) = 5 X(76) - X(7757) = 17 X(2) - 15 X(7786) = 17 X(76) - 5 X(7786) = 3 X(7810) - 2 X(8354) = 10 X(7786) - 17 X(9466) = 8 X(6683) - 13 X(9466) = 2 X(194) - 7 X(9466) = 4 X(3934) - 5 X(9466) = 2 X(7757) - 5 X(9466) = 2 X(2) - 3 X(9466) = 9 X(194) - 7 X(11055) = 18 X(3934) - 5 X(11055) = 9 X(7757) - 5 X(11055) = 9 X(39) - 4 X(11055) = 9 X(9466) - 2 X(11055) = 9 X(76) - X(11055) = X(5188) + 2 X(13108) = X(3534) + 3 X(13108) = X(9887) - 3 X(14568)

X(14711) lies on these lines: {2,39}, {85,7230}, {148,7848}, {183,8589}, {187,8667}, {312,4403}, {511,3830}, {524,14537}, {543,8353}, {574,8556}, {599,11648}, {726,4745}, {732,8584}, {1003,7751}, {2482,13468}, {2782,8703}, {3534,5188}, {3734,5008}, {3845,6248}, {3906,8029}, {5007,11286}, {7603,7813}, {7755,8368}, {7780,13586}, {7805,12150}, {7810,8354}, {7845,11185}

X(14711) = complement of X(11055)
X(14711) = reflection of X(i) in X(j) for these {i,j}: {39, 9466}, {7757, 3934}, {9466, 76}

X(14712) =  ANTICOMPLEMENT OF X(316)

Barycentrics    3 a^4-a^2 b^2-b^4-a^2 c^2+b^2 c^2-c^4 : :

X(14712) lies on the cubic K103 and these lines: {2,187}, {3,7777}, {4,2080}, {6,7833}, {8,5184}, {15,622}, {16,621}, {20,185}, {23,7665}, {30,148}, {32,6655}, {69,5104}, {76,6658}, {83,7830}, {99,754}, {115,12191}, {141,384}, {147,11676}, {183,11361}, {192,4302}, {230,14041}, {315,3552}, {325,13586}, {330,4299}, {376,7774}, {390,5148}, {401,3580}, {524,8591}, {530,3180}, {531,3181}, {532,6780}, {533,6779}, {550,7762}, {574,7812}, {620,7809}, {691,5189}, {736,8782}, {1003,3314}, {1078,7747}, {1326,4201}, {1379,2542}, {1380,2543}, {1384,7806}, {1657,7754}, {1691,3618}, {1975,7893}, {1992,8586}, {2031,5304}, {2458,10350}, {2459,11293}, {2460,9540}, {2549,7766}, {3053,5025}, {3090,14693}, {3091,13449}, {3329,8356}, {3534,7837}, {3600,5194}, {3734,7811}, {3767,10631}, {3785,14035}, {3788,7860}, {3926,7946}, {4045,12150}, {5007,7847}, {5008,7827}, {5013,7921}, {5023,7773}, {5059,6392}, {5103,7876}, {5111,7738}, {5140,6995}, {5149,9866}, {5167,11673}, {5206,7752}, {5291,6653}, {5585,11184}, {6179,7748}, {6390,7840}, {6567,9543}, {6636,8878}, {6656,10583}, {6680,7911}, {7745,7824}, {7756,7760}, {7759,7782}, {7763,7900}, {7768,7816}, {7769,7843}, {7770,7904}, {7775,8588}, {7776,7891}, {7781,7877}, {7784,7892}, {7789,7939}, {7792,7924}, {7795,7929}, {7799,7845}, {7801,7850}, {7807,7885}, {7818,7835}, {7819,7928}, {7820,7883}, {7822,7936}, {7825,7857}, {7828,7842}, {7832,7873}, {7834,7910}, {7846,7935}, {7863,7917}, {7868,14036}, {7875,11287}, {7923,8357}, {7931,8369}, {7938,14001}, {8352,8859}, {8596,11054}, {9218,9514}, {9889,11646}, {13881,14062}

X(14712) = anticomplement X(316)
X(14712) = reflection of X(i) in X(j) for these {i,j}: {4, 2080}, {8, 5184}, {69, 5104}, {99, 6781}, {147, 11676}, {148, 385}, {315, 5162}, {316, 187}, {621, 16}, {622, 15}, {5189, 691}, {5207, 2076}, {7779, 99}, {7840, 8598}, {8591, 9855}, {8596, 11054}
X(14712) = X(67)-Ceva conjugate of X(2)
X(14712) = cevapoint of X(2896) and X(8591)
X(14712) = crossdifference of every pair of points on line {2451, 10567}
X(14712) = polar conjugate of isogonal conjugate of X(23164)
X(14712) = orthoptic-circle-of-Steiner-inellipe-inverse of X(10173)
X(14712) = orthoptic-circle-of-Steiner-circumellipe-inverse of X(6032)
X(14712) = DeLongchamps-circle-inverse of X(12220)
X(14712) = psi-transform of X(10160)
X(14712) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {31, 11061}, {67, 6327}, {2157, 69}, {3455, 8}
X(14712) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 7823, 7785), (32, 6655, 7797), (32, 7802, 6655), (32, 7872, 7856), (187, 316, 2), (315, 3552, 7836), (384, 7750, 2896), (550, 7762, 7783), (1384, 7841, 7806), (3972, 7761, 2), (3972, 11057, 7761), (5023, 7773, 7907), (5475, 7771, 2), (7762, 7783, 13571), (7804, 7831, 2)






leftri  Centers of common circumconics of two triangles: X(14713) - X(14781)  rightri

This preamble and centers X(14713)-X(14781) were contributed by César Eliud Lozada, October 10, 2017.

The following table shows the ETC indexes of centers of conics circumscribing two or more central triangles. For definitions of triangles see the Index of triangles referenced in ETC.

Notes: Centers with complicated coordinates have not been calculated and are marked with "--".
An asterisk * indicates that ABC is autopolar with respect to the conic.

Triangles Center Triangles Center Triangles Center
ABC, Aries 14713 anticomplementary, tangential * 4577 incentral, symmedial 14751
ABC, inner-Conway, Honsberger 9 anticomplementary, X-parabola-tangential * 14728 incentral, Yff contact 14752
ABC, outer-Garcia 10 anticomplementary, X3-ABC reflections 5 incircle-circles, inverse-in-incircle 5045
ABC, Gossard 402 Aquila, excentral (Bevan circle) 40 intangents, 5th mixtilinear 14753
ABC, intangents 14714 Ara, Steiner 14729 intangents, tangential-midarc --
ABC, Johnson (Johnson circumconic) 5 1st Auriga, 2nd Auriga 55 intangents, 2nd tangential-midarc --
ABC, 5th mixtilinear 1 1st Brocard, 2nd Brocard (Brocard circle) 182 intouch, Lemoine 14754
ABC, 1st Schiffler, 2nd Schiffler (Feuerbach hyperbola) 11 4th Brocard, orthocentroidal (orthocentroidal circle) 381 intouch, Macbeath 14755
ABC, tangential-midarc 10493 circumsymmedial, 2nd Ehrmann 14730 intouch, medial 142
ABC, 2nd tangential-midarc 10494 2nd Conway, Steiner 14731 intouch, orthic 14756
anti-Aquila, incentral, medial 1125 2nd Conway, Yff contact 14732 intouch, Steiner 14757
anti-Ara, 2nd anti-Conway 9969 2nd Ehrmann, Kosnita 575 intouch, symmedial 14758
anti-Ara, 2nd Hyacinth 14715 Euler, 2nd Euler, 3rd Euler, 4th Euler, 5th Euler, Feuerbach, medial, orthic (nine-points circle) 5 intouch, Yff contact 14759
anti-Ara, Pelletier -- excenters-midpoints, medial 3035 inverse-in-incircle, Pelletier 14760
anti-Ara, Schroeter -- excenters-reflections, excentral 1 1st Johnson-Yff, inner-Yff 495
1st anti-circumperp, Aries 159 excentral, hexyl 3 1st Johnson-Yff, outer-Yff 496
1st anti-circumperp, inner-Conway 14716 excentral, Pelletier * 14733 Kosnita, Trinh 3
2nd anti-Conway, 2nd Hyacinth 389 excentral, Schroeter * 476 Lemoine, Macbeath 14761
2nd anti-Conway, midheight, Schroeter 11746 excentral, Stammler 110 Lemoine, medial 14762
2nd anti-Conway, Pelletier 14717 excentral, tangential *(Stammler hyperbola) 110 Lemoine, orthic 14763
anti-Euler, 4th anti-Euler 1298 excentral, X-parabola-tangential * 14734 Lemoine, Steiner 14764
anti-Euler, anti-Hutson intouch 14718 extouch, 2nd Hatzipolakis -- Lemoine, symmedial 14765
anti-Euler, anticomplementary 3 extouch, incentral 14735 Lemoine, Yff contact 14766
anti-Euler, hexyl 98 extouch, intouch 5452 Macbeath, medial 14767
3rd anti-Euler, 4th anti-Euler 3 extouch, Lemoine 14736 Macbeath, orthic 14768
3rd anti-Euler, anticomplementary 1303 extouch, Macbeath -- Macbeath, Steiner 14769
3rd anti-Euler, excentral * 14719 extouch, Mandart-incircle(Mandart inellipse) 9 Macbeath, symmedial 14770
3rd anti-Euler, Stammler -- extouch, medial 10 Macbeath, Yff contact 14771
4th anti-Euler, hexyl 14720 extouch, orthic 14737 medial, Steiner 620
anti-excenters-reflections, orthic 4 extouch, Steiner 14738 medial, symmedial 3589
anti-Hutson intouch, anti-Mandart-incircle -- extouch, symmedial 14739 medial, Yff contact 4422
anti-Hutson intouch, hexyl 74 extouch, Yff contact 14740 5th mixtilinear, tangential-midarc --
anti-Hutson intouch, Stammler -- Feuerbach, Lemoine 14741 5th mixtilinear, 2nd tangential-midarc --
anti-Hutson intouch, tangential 3 2nd Hatzipolakis, incentral -- 6th mixtilinear, Yff contact --
anti-incircle-circles, anti-inverse-in-incircle 5 2nd Hatzipolakis, intouch 14742 orthic, Steiner 14772
anti-incircle-circles, Ara (A circle) 7387 2nd Hatzipolakis, Lemoine -- orthic, symmedial 14773
anti-inverse-in-incircle, 2nd Conway 3448 2nd Hatzipolakis, Macbeath -- orthic, Yff contact 14774
anti-inverse-in-incircle, Steiner 14721 2nd Hatzipolakis, medial 14743 1st orthosymmedial, 2nd orthosymmedial 5480
anti-inverse-in-incircle, Yff contact -- 2nd Hatzipolakis, orthic 14744 1st Parry, 2nd Parry, 3rd Parry (Parry circle) 351
anti-Mandart-incircle, excentral 14722 2nd Hatzipolakis, Steiner -- Pelletier, Schroeter * 14775
anti-Mandart-incircle, Stammler 100 2nd Hatzipolakis, symmedial -- Pelletier, tangential * 14776
anti-Mandart-incircle, tangential 14723 2nd Hatzipolakis, Yff contact Pelletier, X-parabola-tangential * 14777
anti-McCay, 1st Parry 14724 Hutson intouch, intouch, Mandart-incircle, midarc, 2nd midarc (incircle) 1 Schroeter, tangential * 685
6th anti-mixtilinear, midheight 14725 2nd Hyacinth, Pelletier -- Schroeter, X-parabola-tangential * 5466
6th anti-mixtilinear, Schroeter 14726 2nd Hyacinth, Schroeter 14745 Stammler, tangential 110
anticomplementary, Aquila 10 incentral, intouch 14746 Stammler, X3-ABC reflections (Stammler circle) 3
anticomplementary, excentral * 99 incentral, Lemoine 14747 Steiner, symmedial 14778
anticomplementary, Pelletier * 14727 incentral, Macbeath 14748 Steiner, Yff contact 14779
anticomplementary, Schroeter * 892 incentral, Mandart-incircle, orthic 14749 symmedial, Yff contact 14780
anticomplementary, Stammler 930 incentral, Steiner 14750 tangential, X-parabola-tangential * 14781

Triangles circumscribed by the circumcircle of ABC: ABC, ABC-X3 reflections, 1st anti-circumperp, circummedial, circumnormal, circumorthic, 1st circumperp, 2nd circumperp, circumsymmedial, circumtangential, 3rd mixtilinear, 4th mixtilinear.

underbar

X(14713) = CENTER OF THE {ABC, ARIES}-CIRCUMCONIC

Barycentrics    a^2*(a^4+(b^2-c^2)^2)*(a^6-(b^2+c^2)*a^4+(b^2+c^2)^2*a^2-(b^2+c^2)*(b^4+c^4)) : :

X(14713) lies on these lines: {2,157}, {22,5976}, {25,14715}, {39,184}, {159,6503}, {3155,13882}, {5938,11165}, {6337,11206}

X(14714) = CENTER OF THE {ABC, INTANGENTS}-CIRCUMCONIC

Barycentrics    a^2*(b-c)^2*(a^4-(b+c)*a^3-(b^2-b*c+c^2)*a^2+(b+c)*(b^2+c^2)*a-b*c*(b-c)^2)*(-a+b+c)^2 : :

The {ABC,intangents}-circumconic is the isogonal conjugate of the Soddy line. Let P and U be the circumcircle intercepts of the Soddy line. Then X(14714) is the crosssum of P and U. (Randy Hutson, November 2, 2017)

Let A'B'C' be the orthic triangle. Let La be the Soddy line of triangle AB'C', and define Lb and Lc cyclically. Let A" = Lb∩Lc, and define B" and C" cyclically. Triangle A"B"C" is inversely similar to ABC, with similicenter X(14714). (Randy Hutson, July 31 2018)

X(14714) = X(2)-Ceva conjugate of X(657)

p> X(14714) lies on these lines: {1,4566}, {11,3138}, {55,14723}, {244,9511}, {663,3022}, {997,4319}, {1015,3269}, {1042,12262}, {1064,2293}, {1458,11714}, {1984,2310}

X(14715) = CENTER OF THE {ANTI-ARA, 2nd HYACINTH}-CIRCUMCONIC

Barycentrics    ((b^2+c^2)*a^6-(b^2+c^2)^2*a^4+(b^4-c^4)*(b^2-c^2)*a^2-(b^4+c^4)*(b^2-c^2)^2)*(a^2-b^2+c^2)*(a^2+b^2-c^2) : :

X(14715) lies on these lines: {5,14768}, {25,14713}, {53,428}, {136,1196}, {427,2023}, {2501,13567}, {5254,11245}

X(14716) = CENTER OF THE {1st ANTI-CIRCUMPERP, INNER-CONWAY}-CIRCUMCONIC

Barycentrics    a^6-(b+c)*a^5-(b^2+c^2)*a^4+2*b*c*(b+c)*a^3+(b^3-c^3)*(b-c)*a^2+(b^3-c^3)*(b^2-c^2)*a-(b^2+b*c+c^2)^2*(b-c)^2 : :

X(14716) lies on these lines: {2,14756}, {319,3681}

X(14716) = anticomplement of X(14756)

X(14717) = CENTER OF THE {2nd ANTI-CONWAY, PELLETIER}-CIRCUMCONIC

Barycentrics    a^2*((b^2+c^2)*a^7-(b+c)*(b^2+c^2)*a^6-(b^4-4*b^2*c^2+c^4)*a^5+(b^2-c^2)*(b-c)*(b^2+4*b*c+c^2)*a^4-(b^4+c^4+2*b*c*(b^2+3*b*c+c^2))*(b-c)^2*a^3+(b^4-c^4)*(b-c)^3*a^2+(b^2-c^2)^2*(b^4+c^4)*a-(b^4-c^4)*(b^2-c^2)^2*(b-c)) : :
X(14717) = 3*X(51)+X(3270)

X(14717) lies on these lines: {6,692}, {33,51}, {389,946}

X(14718) = CENTER OF THE {ANTI-EULER, ANTI-HUTSON INTOUCH}-CIRCUMCONIC

Barycentrics    a^12-2*(b^4-b^2*c^2+c^4)*a^8+(3*b^8+3*c^8-b^2*c^2*(2*b^4+b^2*c^2+2*c^4))*a^4-2*(b^4-c^4)^2*(b^2+c^2)*a^2-(b^2-c^2)^2*b^4*c^4 : :

X(14718) lies on these lines: {3,4577}, {98,9479}, {12117,12122}

X(14718) = reflection of X(4577) in X(3)

X(14719) = CENTER OF THE {3rd ANTI-EULER, EXCENTRAL}-CIRCUMCONIC

Barycentrics    a^2*f(b,c,a)*f(c,a,b) : : , where f(a,b,c)=(a^2-b*(b+c))*(a^2-b*(b-c))*(a^2-c*(b+c))*(a^2+c*(b-c))*(b^2-c^2)

X(14719) lies on the circumcircle and the line {3,14720}

X(14719) = reflection of X(14720) in X(3)
X(14719) = trilinear product of vertices of circumorthic-of-circumorthic triangle
X(14719) = antipode of X(14720) in circumcircle

X(14720) = CENTER OF THE {4th ANTI-EULER, HEXYL}-CIRCUMCONIC

Barycentrics    a^2*f(b,c,a)*f(c,a,b) : : , where f(a,b,c)=(b^2+c^2)*a^10-(3*b^4+2*b^2*c^2+3*c^4)*a^8+(b^2+c^2)*(3*b^4-2*b^2*c^2+3*c^4)*a^6-(b^8+c^8)*a^4+(b^4-c^4)*(b^2-c^2)*b^2*c^2*a^2-(b^2-c^2)^4*b^2*c^2

X(14720) lies on the circumcircle and these lines: {3,14719}, {184,12092}

X(14720) = reflection of X(14719) in X(3)
X(14720) = antipode of X(14719) in circumcircle

X(14721) = CENTER OF THE {ANTI-INVERSE-IN-INCIRCLE, STEINER}-CIRCUMCONIC

Barycentrics    a^16-2*(b^2+c^2)*a^14+(b^4+4*b^2*c^2+c^4)*a^12+2*(b^4-3*b^2*c^2+c^4)*(b^2+c^2)*a^10-(6*b^8+6*c^8-b^2*c^2*(6*b^4+b^2*c^2+6*c^4))*a^8+6*(b^8-c^8)*(b^2-c^2)*a^6-(b^2-c^2)^2*(3*b^8+3*c^8+2*b^2*c^2*(b^4+c^4))*a^4+2*(b^6-c^6)*(b^2-c^2)^2*(b^4-c^4)*a^2-(b^8+c^8+b^2*c^2*(2*b^4+b^2*c^2+2*c^4))*(b^2-c^2)^4 : :

X(14721) lies on these lines: {2,685}, {147,858}, {401,1503}

X(14721) = anticomplement of X(685)

X(14722) = CENTER OF THE {ANTI-MANDART-INCIRCLE, EXCENTRAL}-CIRCUMCONIC

Barycentrics    a*(a-b)*(a-c)*(a^5-3*(b+c)*a^4+3*(b^2+b*c+c^2)*a^3-(2*b^2+b*c+2*c^2)*(b-c)^2*a+(b^2-c^2)*(b-c)^3) : :

X(14722) lies on these lines: {}

X(14723) = CENTER OF THE {ANTI-MANDART-INCIRCLE, TANGENTIAL}-CIRCUMCONIC

Barycentrics    a^2*(a-b)*(a-c)*((b-c)^2*a^4-(b^2-c^2)*(b-c)*a^3-(b^4+c^4-3*b*c*(b^2+c^2))*a^2+(b^4-c^4)*(b-c)*a-(b^2-c^2)^2*b*c) : :

X(14723) lies on these lines: {55,14714}, {100,1624}

X(14724) = CENTER OF THE {ANTI-MCCAY, 1st PARRY}-CIRCUMCONIC

Barycentrics    a^12-2*(b^2+c^2)*a^10-(23*b^4-51*b^2*c^2+23*c^4)*a^8+32*(b^4-c^4)*(b^2-c^2)*a^6-(17*b^8+17*c^8-b^2*c^2*(19*b^4-9*b^2*c^2+19*c^4))*a^4+2*(b^2+c^2)*(5*b^8+5*c^8-b^2*c^2*(20*b^4-31*b^2*c^2+20*c^4))*a^2-(2*b^2-c^2)*(b^2-2*c^2)*(b^8+c^8+b^2*c^2*(2*b^4-7*b^2*c^2+2*c^4)) : :

X(14724) lies on these lines: {542,9855}, {671,9227}, {2793,14041}

X(14725) = CENTER OF THE {6th ANTI-MIXTILINEAR, MIDHEIGHT}-CIRCUMCONIC

Barycentrics    (b^4+c^4)*a^8-2*(b^6+c^6)*a^6+2*(b^4-c^4)^2*a^4-2*(b^8-c^8)*a^2*(b^2-c^2)+(b^2-c^2)^2*(b^4+c^4)^2 : :

X(14725) lies on these lines: {2,11610}, {626,3819}

X(14726) = CENTER OF THE {6th ANTI-MIXTILINEAR, SCHROETER}-CIRCUMCONIC

Barycentrics    (b^2+c^2)*a^6-(b^4+6*b^2*c^2+c^4)*a^4-(b^2+c^2)*(2*b^4-9*b^2*c^2+2*c^4)*a^2+((b^2-c^2)^2-4*b^2*c^2)*b^2*c^2 : :

X(14726) lies on these lines: {2,14772}, {76,338}, {1084,13468}

X(14726) = complement of X(14772)

X(14727) = CENTER OF THE {ANTICOMPLEMENTARY, PELLETIER}-CIRCUMCONIC

Barycentrics    f(b,c,a)*f(c,a,b) : : , where f(a,b,c)=a*(b-c)*(a^2-(b+c)*a+2*b*c)*((b+c)*a-b^2-c^2)

X(14727) lies on the Steiner circumellipse and these lines: {190,657}, {513,4569}, {663,664}, {668,3900}, {670,7253}, {927,6613}

X(14728) = CENTER OF THE {ANTICOMPLEMENTARY, X-PARABOLA-TANGENTIAL}-CIRCUMCONIC

Barycentrics    f(b,c,a)*f(c,a,b) : : , where f(a,b,c)=(b^2-c^2)*(2*a^2-b^2-c^2)*(2*a^4-2*(b^2+c^2)*a^2+b^4+c^4)

X(14728) lies on the Steiner circumellipse and the line {99,8029}

X(14729) = CENTER OF THE {ARA, STEINER}-CIRCUMCONIC

Barycentrics    a^2*(a^14-3*(b^2+c^2)*a^12+(b^4+12*b^2*c^2+c^4)*a^10+(b^2+c^2)*(5*b^4-18*b^2*c^2+5*c^4)*a^8-(5*b^8-19*b^4*c^4+5*c^8)*a^6-(b^2+c^2)*(b^8+c^8-b^2*c^2*(14*b^4-27*b^2*c^2+14*c^4))*a^4+3*(b^2-c^2)^2*(b^8+c^8-2*(b^4+b^2*c^2+c^4)*b^2*c^2)*a^2+(b^4-c^4)*(b^2-c^2)*(-b^8-c^8+2*b^2*c^2*(b^4+c^4))) : :

X(14729) lies on the tangential circle and these lines: {3,5099}, {22,2770}, {23,5866}, {24,935}, {186,13200}, {468,8428}, {523,2079}, {2070,2936}, {2453,6644}, {2930,2931}, {3566,10117}, {5926,7669}

X(14729) = midpoint of X(23) and X(5866)
X(14729) = circumcircle-inverse-of-X(5099)

X(14730) = CENTER OF THE {CIRCUMSYMMEDIAL, 2nd EHRMANN}-CIRCUMCONIC

Barycentrics    a^2*(7*a^6-16*(b^2+c^2)*a^4-(23*b^4-104*b^2*c^2+23*c^4)*a^2-12*b^2*c^2*(b^2+c^2)) : :

X(14730) lies on these lines: {}

X(14731) = CENTER OF THE {2nd CONWAY, STEINER}-CIRCUMCONIC

Barycentrics    a^12-2*(b^2+c^2)*a^10+3*(b^4+c^4)*a^8-2*(b^2+c^2)*(3*b^4-5*b^2*c^2+3*c^4)*a^6+(5*b^8+5*c^8-b^2*c^2*(b^2+2*c^2)*(2*b^2+c^2))*a^4-4*(b^4-c^4)*(b^2-c^2)*b^2*c^2*a^2-(b^4+c^4)*(b^2-c^2)^4 : :
X(14731) = 3*X(2)-4*X(3258) = 3*X(1138)-X(12383)

The {2nd Conway, Steiner}-circumconic passes through X(3), X(8), and the extraversions of X(8). (Randy Hutson, November 2, 2017)

X(14731) lies on the anticomplementary circle and these lines: {2,476}, {20,477}, {23,12384}, {30,146}, {147,5189}, {325,1272}, {511,11751}, {523,3448}, {2453,5169}, {6563,13219}, {9158,9999}

X(14731) = reflection of X(i) in X(j) for these (i,j): (20, 477), (476, 3258)
X(14731) = isogonal conjugate of X(34191)
X(14731) = anticomplement of X(476)
X(14731) = orthoptic-circle-of-Steiner-circumellipse-inverse of X(842)
X(14731) = de-Longchamps-circle-inverse of X(10420)
X(14731) = anticomplementary-circle antipode of X(34193)
X(14731) = {X(476), X(3258)}-harmonic conjugate of X(2)

X(14732) = CENTER OF THE {2nd CONWAY, YFF CONTACT}-CIRCUMCONIC

Barycentrics    a^8-2*(b+c)*a^7+(b^2+4*b*c+c^2)*a^6+2*(b+c)*(b^2-3*b*c+c^2)*a^5-(6*b^4+6*c^4-b*c*(6*b^2+b*c+6*c^2))*a^4+6*(b^4-c^4)*(b-c)*a^3-(3*b^4+3*c^4+2*b*c*(b^2+c^2))*(b-c)^2*a^2+2*(b^2-c^2)*(b-c)^2*(b^3-c^3)*a-(b^4+c^4+b*c*(2*b^2+b*c+2*c^2))*(b-c)^4 : :
X(14732) = 3*X(2)-4*X(1566)

X(14732) lies on the anticomplementary circle and these lines: {2,927}, {20,2724}, {150,514}, {152,516}

X(14732) = reflection of X(i) in X(j) for these (i,j): (20, 2724), (927, 1566)
X(14732) = anticomplement of X(927)
X(14732) = {X(927), X(1566)}-harmonic conjugate of X(2)

X(14733) = CENTER OF THE {EXCENTRAL, PELLETIER}-CIRCUMCONIC

Barycentrics    a^2*f(b,c,a)*f(c,a,b) : : , where f(a,b,c)=(b-c)*(-a+b+c)*(2*a^2-(b+c)*a-(b-c)^2)

Let A', B', C' be the intersections of line X(2)X(7) and lines BC, CA, AB, resp. The circumcircles of AB'C', BC'A', CA'B' concur in X(14733). (Randy Hutson, November 2, 2017)

X(14733) lies on the circumcircle and these lines: {1,2717}, {36,103}, {56,840}, {59,101}, {99,4620}, {100,3900}, {104,971}, {105,1319}, {106,1458}, {108,7128}, {109,663}, {390,2724}, {513,934}, {517,972}, {651,14074}, {664,9086}, {885,927}, {901,2283}, {953,999}, {1055,2078}, {1121,1311}, {1295,3100}, {1471,12032}, {2688,4304}, {2716,3576}, {2718,13462}, {2723,5731}, {2745,3428}, {3660,10426}, {5061,6015}, {9058,14513}

X(14733) = isogonal conjugate of X(6366)
X(14733) = trilinear pole of the line {6, 109}
X(14733) = Ψ(X(6), X(109))
X(14733) = Ψ(X(7), X(11))
X(14733) = Λ(X(1), X(676))
X(14733) = Λ(X(11), X(1146))
X(14733) = Ψ(X(650), X(1))

X(14734) = CENTER OF THE {EXCENTRAL, X-PARABOLA-TANGENTIAL}-CIRCUMCONIC

Barycentrics    f(b,c,a)*f(c,a,b) : : , where f(a,b,c)=(b^2-c^2)*(((b^2+c^2-a^2)^2-b^2*c^2)^2-b^2*c^2*(2*a^2-b^2-c^2)^2)

X(14734) lies on the circumcircle and these lines: {74,6321}, {110,8029}, {842,6036}

X(14735) = CENTER OF THE {EXTOUCH, INCENTRAL}-CIRCUMCONIC

Barycentrics    a*(-a+b+c)*((b+c)^2*a^4+(b+c)*(b^2-4*b*c+c^2)*a^3-(b^4+c^4+b*c*(b^2-12*b*c+c^2))*a^2-(b^2-c^2)*(b-c)^3*a-(b^2-c^2)^2*b*c) : :

This conic is the bicevian conic of X(1) and X(8).

X(14735) lies on these lines: {210,7069}, {3686,14749}

X(14736) = CENTER OF THE {EXTOUCH, LEMOINE}-CIRCUMCONIC

Barycentrics    (-a+b+c)*(24*a^7-8*(b+c)*a^6-6*(3*b^2-4*b*c+3*c^2)*a^5+(b+c)*(31*b^2-34*b*c+31*c^2)*a^4-3*(5*b^4+5*c^4+2*b*c*(b^2-5*b*c+c^2))*a^3-(b+c)*(19*b^4+19*c^4-b*c*(49*b^2-44*b*c+49*c^2))*a^2+3*(3*b^4+3*c^4+2*b*c*(b^2+c^2))*(b-c)^2*a-(b^4-c^4)*(b-c)*(4*b^2-3*b*c+4*c^2)) : :

This conic is the bicevian conic of X(8) and X(598).

X(14736) lies on these lines: {}

X(14737) = CENTER OF THE {EXTOUCH, ORTHIC}-CIRCUMCONIC

Barycentrics    a*(-a+b+c)*(2*a^6-(b+c)*a^5-2*(b^2-b*c+c^2)*a^4+(b+c)*(3*b^2-4*b*c+3*c^2)*a^3-(b^2+3*b*c+c^2)*(b-c)^2*a^2-2*(b^4-c^4)*(b-c)*a+(b-c)*(b^2-c^2)*(b^3+c^3)) : :

This conic is the bicevian conic of X(4) and X(8).

X(14737) lies on these lines: {44,2262}, {210,212}

X(14738) = CENTER OF THE {EXTOUCH, STEINER}-CIRCUMCONIC

Barycentrics    (-a+b+c)*(2*(b+c)*a^6-2*(b-c)^2*a^5+(b-3*c)*(3*b-c)*(b+c)*a^4-(5*b^2+14*b*c+5*c^2)*(b-c)^2*a^3-(b+c)*(b^4+c^4-b*c*(9*b^2-14*b*c+9*c^2))*a^2+(3*b^4+3*c^4+4*b*c*(b^2+c^2))*(b-c)^2*a-(b^2-c^2)*(b-c)^3*b*c) : :

This conic is the bicevian conic of X(8) and X(99).

X(14738) lies on these lines: {}

X(14739) = CENTER OF THE {EXTOUCH, SYMMEDIAL}-CIRCUMCONIC

Barycentrics    a^2*(-a+b+c)*((b^2+c^2)^2*a^5+(b^3+c^3)*(b-c)^2*a^4-(b^6+c^6+(3*b^2-4*b*c+3*c^2)*b^2*c^2)*a^3-(b+c)*(b^6+c^6-(3*b^4+3*c^4-4*b*c*(b-c)^2)*b*c)*a^2+2*(b^3-c^3)*(b-c)*b^2*c^2*a-(b^4-c^4)*b^2*c^2*(b-c)) : :

This conic is the bicevian conic of X(6) and X(8).

X(14739) lies on these lines: {}

X(14740) = CENTER OF THE {EXTOUCH, YFF CONTACT}-CIRCUMCONIC

Barycentrics    a*(-a+b+c)*((b+c)*a^3-(b+c)^2*a^2-(b+c)*(b^2-3*b*c+c^2)*a+(b^3-c^3)*(b-c)) : :
X(14740) = X(11)-3*X(210) = X(100)+3*X(3681) = X(1320)-5*X(3876) = 8*X(3678)-X(9951) = 3*X(3740)-2*X(6667) = 5*X(4005)+X(13996) = 3*X(5692)-X(12758)

This conic is the Mandart hyperbola and the bicevian conic of X(8) and X(190).

X(14740) lies on these lines: {8,80}, {10,12736}, {11,210}, {40,12059}, {63,100}, {72,1145}, {78,214}, {153,5815}, {480,6594}, {518,3035}, {758,6735}, {952,6737}, {956,11715}, {960,5854}, {1320,3876}, {1387,5044}, {2829,12527}, {3036,4662}, {3059,6068}, {3421,12751}, {3740,6667}, {3874,5552}, {3877,8275}, {3927,12515}, {3940,6265}, {4005,13996}, {4015,6702}, {4420,4996}, {4863,13274}, {4882,5541}, {4915,12653}, {5082,14217}, {5220,13205}, {5248,7162}, {5250,13278}, {5904,7080}, {9954,13227}

X(14740) = midpoint of X(i) and X(j) for these {i,j}: {40, 12665}, {72, 1145}, {3059, 6068}, {5904, 11570}
X(14740) = reflection of X(i) in X(j) for these (i,j): (1387, 5044), (3036, 4662), (5083, 3035), (6702, 4015), (12736, 10)
X(14740) = excentral-to-ABC barycentric image of X(11)

X(14741) = CENTER OF THE {FEUERBACH, LEMOINE}-CIRCUMCONIC

Barycentrics    4*(b^2+c^2)*a^10-8*(3*b^4+7*b^2*c^2+3*c^4)*a^8+(b^2+c^2)*(53*b^4-19*b^2*c^2+53*c^4)*a^6-(53*b^8+53*c^8-2*b^2*c^2*(23*b^4+54*b^2*c^2+23*c^4))*a^4+(b^2+c^2)*(24*b^8+24*c^8-b^2*c^2*(125*b^4-197*b^2*c^2+125*c^4))*a^2-(b^2-c^2)^2*(b^2-2*c^2)^2*(2*b^2-c^2)^2 : :

X(14741) lies on the line {5,598}

X(14742) = CENTER OF THE {2nd HATZIPOLAKIS, INTOUCH}-CIRCUMCONIC

Barycentrics    a*(a-b+c)*(a+b-c)*(2*a^7-3*(b+c)*a^6+(b^2+c^2)*a^5+(b+c)*(5*b^2-6*b*c+5*c^2)*a^4-(6*b^4+6*c^4-b*c*(b+c)^2)*a^3+(b^2-c^2)*(b-c)*(b^2+5*b*c+c^2)*a^2+(b^2-c^2)^2*(3*b^2-b*c+3*c^2)*a-3*(b^3+c^3)*(b^2-c^2)^2) : :

X(14742) lies on the line {65,603}

X(14743) = CENTER OF THE {2nd HATZIPOLAKIS, MEDIAL}-CIRCUMCONIC

Barycentrics    2*(b+c)*a^7-(3*b^2-4*b*c+3*c^2)*a^6-2*(b+c)*(b^2+c^2)*a^5+(5*b^4+5*c^4-2*b*c*(2*b^2-3*b*c+2*c^2))*a^4-2*(b^2-c^2)^2*(b+c)*a^3-(b^2-c^2)^2*(b^2+4*b*c+c^2)*a^2+2*(b^4-c^4)*(b^2-c^2)*(b+c)*a-(b^2-c^2)^2*(b-c)^4 : :
X(14743) = 3*X(2)+X(1119)

X(14743) lies on these lines: {2,1119}, {5,142}

X(14744) = CENTER OF THE {2nd HATZIPOLAKIS, ORTHIC}-CIRCUMCONIC

Barycentrics    a*(a^2+b^2-c^2)*(a^2-b^2+c^2)*(a^7-(b+c)*a^6+(b-c)^2*a^5+(b^2+c^2)*(b+c)*a^4-(b^4+c^4+2*b*c*(b^2-b*c+c^2))*a^3+(b^4-c^4)*(b-c)*a^2-(b^4+c^4-2*b*c*(b^2+b*c+c^2))*(b-c)^2*a-(b^4-c^4)*(b^2+c^2)*(b-c)) : :

X(14744) lies on the line {19,614}

X(14745) = CENTER OF THE {2nd HYACINTH, SCHROETER}-CIRCUMCONIC

Barycentrics    ((b^2-c^2)^2-4*b^2*c^2)*a^12-(b^2+c^2)*(3*b^4-14*b^2*c^2+3*c^4)*a^10+(2*b^8+2*c^8-b^2*c^2*(b^4+22*b^2*c^2+c^4))*a^8+2*(b^2+c^2)*(b^8+c^8-6*(b^2-c^2)^2*b^2*c^2)*a^6-(b^4-c^4)^2*(3*b^4-8*b^2*c^2+3*c^4)*a^4+(b^4-c^4)^3*(b^2-c^2)*a^2-(b^2-c^2)^6*b^2*c^2 : :

X(14745) lies on these lines: {570,3018}, {3269,5254}

X(14746) = CENTER OF THE {INCENTRAL, INTOUCH}-CIRCUMCONIC

Barycentrics    a*((b+c)^2*a^3-2*(b+c)*(b^2+b*c+c^2)*a^2+(b^2+3*b*c+c^2)*(b-c)^2*a+(b^2-c^2)*(b-c)*b*c) : :

This conic is the bicevian conic of X(1) and X(7).

X(14746) lies on these lines: {37,38}, {241,553}, {650,13405}

X(14747) = CENTER OF THE {INCENTRAL, LEMOINE}-CIRCUMCONIC

Barycentrics    a*(2*(3*b^2+2*b*c+3*c^2)*a^5-16*b*c*(b+c)*a^4-(9*b^4+9*c^4+b*c*(25*b^2-12*b*c+25*c^2))*a^3-2*b*c*(b+c)*(b^2+6*b*c+c^2)*a^2+(b^2-3*b*c+c^2)*(3*b^4+3*c^4+b*c*(7*b^2+12*b*c+7*c^2))*a+(b+c)*(5*b^4+5*c^4-b*c*(3*b^2+8*b*c+3*c^2))*b*c) : :

This conic is the bicevian conic of X(1) and X(598).

X(14747) lies on these lines: {}

X(14748) = CENTER OF THE {INCENTRAL, MACBEATH}-CIRCUMCONIC

Barycentrics    a*((b^4+c^4+3*b*c*(b^2+c^2))*a^7-(b^6+c^6+2*b*c*(3*b^4+b^2*c^2+3*c^4))*a^5-(b+c)*(b^4+c^4-b*c*(b^2-4*b*c+c^2))*b*c*a^4-(b^2-c^2)^2*(b^4+c^4-3*b*c*(b^2+c^2))*a^3+2*(b^2-c^2)*(b-c)*(b^4+c^4+b*c*(b^2+b*c+c^2))*b*c*a^2+(b^2-c^2)^2*(b+c)^2*(b^2-b*c+c^2)^2*a-(b^2-c^2)^2*(b+c)*(b^4+c^4-b*c*(b+c)^2)*b*c) : :

This conic is the bicevian conic of X(1) and X(264).

X(14748) lies on these lines: {}

X(14749) = CENTER OF THE {INCENTRAL, MANDART-INCIRCLE, ORTHIC}-CIRCUMCONIC

Barycentrics    a*(-a+b+c)*((b+c)^2*a^4+(b+c)*(b^2+c^2)*a^3-(b^2+3*b*c+c^2)*(b-c)^2*a^2-(b^2-c^2)^2*(b+c)*a-(b^2-c^2)^2*b*c) : :

This conic is the bicevian conic of X(1) and X(4).

X(14749) lies on these lines: {11,2092}, {37,1953}, {478,11496}, {497,941}, {950,5724}, {1100,11998}, {1839,1841}, {2277,11376}, {2310,2667}, {3686,14735}, {3706,3965}, {3731,9898}, {4343,14100}

X(14750) = CENTER OF THE {INCENTRAL, STEINER}-CIRCUMCONIC

Barycentrics    a*(b+c)*((b+c)*a^5+2*b*c*a^4+(b+c)*(b^2-4*b*c+c^2)*a^3-b*c*(3*b^2-2*b*c+3*c^2)*a^2+b*c*(b+c)*(3*b^2-5*b*c+3*c^2)*a-b^2*c^2*(b^2-4*b*c+c^2)) : :

This conic is the bicevian conic of X(1) and X(99).

X(14750) lies on these lines: {}

X(14751) = CENTER OF THE {INCENTRAL, SYMMEDIAL}-CIRCUMCONIC

Barycentrics    a^2*((b-c)^2*a^2-(b+c)^3*a-b*c*(3*b^2+2*b*c+3*c^2)) : :

This conic is the bicevian conic of X(1) and X(6).

X(14751) lies on these lines: {649,3666}, {672,1100}, {1107,3989}, {1500,11205}, {2308,3747}

X(14752) = CENTER OF THE {INCENTRAL, YFF CONTACT}-CIRCUMCONIC

Barycentrics    a*(b+c)*((b+c)*a^3+(b^2-4*b*c+c^2)*a^2-b*c*(b+c)*a+2*b^2*c^2) : :
X(14752) = X(244)-3*X(1962)

This conic is the bicevian conic of X(1) and X(190).

X(14752) lies on these lines: {1,4427}, {42,3952}, {244,1962}, {659,3722}, {740,899}, {1193,4065}, {2292,2611}, {2667,4117}, {2802,3743}, {4432,8054}

X(14752) = complement wrt incentral triangle of X(244)

X(14753) = CENTER OF THE {INTANGENTS, 5th MIXTILINEAR}-CIRCUMCONIC

Barycentrics    a*(-a+b+c)*((b+c)^2*a^6+2*(b^3+c^3)*a^5-b*c*(b+c)^2*a^4-(b^2-c^2)*(b-c)*(2*b^2-b*c+2*c^2)*a^3-(b^2+c^2)*(b^2+3*b*c+c^2)*(b-c)^2*a^2-(b^2-c^2)*(b-c)*b*c*(3*b^2+2*b*c+3*c^2)*a-2*(b^2-c^2)^2*b^2*c^2) : :

X(14753) lies on these lines: {1,9551}, {38,3057}

X(14754) = CENTER OF THE {INTOUCH, LEMOINE}-CIRCUMCONIC

Barycentrics    24*a^6-16*(b+c)*a^5-2*(b^2+4*b*c+c^2)*a^4-(b+c)*(29*b^2-42*b*c+29*c^2)*a^3+2*(7*b^4+7*c^4-b*c*(11*b^2-2*b*c+11*c^2))*a^2+(b+c)*(5*b^4+5*c^4-b*c*(3*b^2+8*b*c+3*c^2))*a+(b^2+c^2)*(4*b^2+3*b*c+4*c^2)*(b-c)^2 : :

This conic is the bicevian conic of X(7) and X(598).

X(14754) lies on these lines: {}

X(14755) = CENTER OF THE {INTOUCH, MACBEATH}-CIRCUMCONIC

Barycentrics    2*(b^2+c^2)*a^8+(b^2-c^2)*(b-c)*a^7-2*(2*b^4+2*c^4-b*c*(b^2+c^2))*a^6-(b+c)*(b^4+c^4-b*c*(b^2-4*b*c+c^2))*a^5+(2*b^4+2*c^4-b*c*(b^2+c^2))*(b-c)^2*a^4-(b^2-c^2)*(b-c)*(b^4+c^4-2*b*c*(b+c)^2)*a^3+2*(b^2-c^2)^2*b*c*(2*b^2-3*b*c+2*c^2)*a^2+(b^2-c^2)^2*(b+c)*(b^4+c^4-3*b*c*(b-c)^2)*a-(b^2-c^2)^4*b*c : :

This conic is the bicevian conic of X(7) and X(264).

X(14755) lies on these lines: {}

X(14756) = CENTER OF THE {INTOUCH, ORTHIC}-CIRCUMCONIC

Barycentrics    a*(2*a^5-(b+c)*a^4-(b^2+c^2)*a^3-2*(b^2-c^2)*(b-c)*a^2+(b^3-c^3)*(b-c)*a+(b^2-c^2)*(b^3-c^3)) : :

This conic is the bicevian conic of X(4) and X(7).

X(14756) lies on these lines: {2,14716}, {48,354}

X(14756) = complement of X(14716)

X(14757) = CENTER OF THE {INTOUCH, STEINER}-CIRCUMCONIC

Barycentrics    (b+c)*(2*a^5+(3*b-c)*(b-3*c)*a^3+2*(b^2-c^2)*(b-c)*a^2-(3*b^4+3*c^4-b*c*(5*b^2-2*b*c+5*c^2))*a+(b^2-c^2)*(b-c)*b*c) : :

This conic is the bicevian conic of X(7) and X(99).

X(14757) lies on these lines: {}

X(14758) = CENTER OF THE {INTOUCH, SYMMEDIAL}-CIRCUMCONIC

Barycentrics    a^2*((b^2+c^2)^2*a^4-(b+c)*(2*b^2-b*c+c^2)*(b^2-b*c+2*c^2)*a^3+(b^2+c^2)*(b^4+c^4-b*c*(b+c)^2)*a^2+(b^2-c^2)*(b-c)*b^2*c^2*a+b^2*c^2*(b^2+c^2)*(b-c)^2) : :

This conic is the bicevian conic of X(6) and X(7).

X(14758) lies on these lines: {}

X(14759) = CENTER OF THE {INTOUCH, YFF CONTACT}-CIRCUMCONIC

Barycentrics    (b+c)*a^3+(3*b-c)*(b-3*c)*a^2-(2*b-c)*(b-2*c)*(b+c)*a+b*c*(b-c)^2 : :

This conic is the bicevian conic of X(7) and X(190). (Randy Hutson, June 27, 2018)

X(14759) lies on these lines: {1,3732}, {192,537}, {1015,3752}

X(14760) = CENTER OF THE {INVERSE-IN-INCIRCLE, PELLETIER}-CIRCUMCONIC

Barycentrics    a*((b+c)*a^5-(3*b^2-2*b*c+3*c^2)*a^4+(b+c)*(4*b^2-7*b*c+4*c^2)*a^3-(4*b^2+9*b*c+4*c^2)*(b-c)^2*a^2+(b^2-c^2)*(b-c)*(3*b^2+b*c+3*c^2)*a-(b^3-c^3)*(b-c)^3) : :
X(14760) = 3*X(354)+X(3022) X(14760) lies on these lines: {1,41}, {103,3333}, {116,11019}, {150,10580}, {152,11037}, {354,3022}, {999,2823}, {1387,2801}, {2784,6744}, {2808,5045}, {6710,13405}, {10697,11529}, {14100,14519}

X(14760) = midpoint of X(1) and X(11028)
X(14760) = incircle-inverse-of-X(5540)

X(14761) = CENTER OF THE {LEMOINE, MACBEATH}-CIRCUMCONIC

Barycentrics    4*(b^2+c^2)*a^10-(3*b^4-2*b^2*c^2+3*c^4)*a^8-(b^2+c^2)*(5*b^4+4*b^2*c^2+5*c^4)*a^6+(3*b^8+3*c^8-2*b^2*c^2*(b^2+2*c^2)*(2*b^2+c^2))*a^4+(b^4-c^4)*(b^2-c^2)*(b^4+9*b^2*c^2+c^4)*a^2-(b^2-c^2)^2*b^2*c^2*(b^2-3*b*c+c^2)*(b^2+3*b*c+c^2) : :

This conic is the bicevian conic of X(264) and X(598).

X(14761) lies on the line {858,3613}

X(14762) = CENTER OF THE {LEMOINE, MEDIAL}-CIRCUMCONIC

Barycentrics    4*a^4+11*(b^2+c^2)*a^2-2*(b^2+3*b*c+c^2)*(b^2-3*b*c+c^2) : :
X(14762) = 3*X(2)+X(598) = 13*X(2)-X(11057) = 13*X(598)+3*X(11057) = 7*X(3934)+2*X(7838) = X(3934)-4*X(8367) = 10*X(6683)-X(7756) = 2*X(6683)+X(8370) = X(7756)+5*X(8370) = X(7838)+14*X(8367) = X(8592)+3*X(9166)

This conic is the bicevian conic of X(2) and X(598).

X(14762) lies on these lines: {2,187}, {5,10168}, {524,3934}, {543,2023}, {574,11164}, {3589,5461}, {3734,11165}, {5032,7805}, {5038,7617}, {5485,14482}, {6683,7756}, {7618,7816}, {7775,7849}, {7840,10302}, {7880,11184}, {8592,9166}

X(14763) = CENTER OF THE {LEMOINE, ORTHIC}-CIRCUMCONIC

Barycentrics    12*a^8-11*(b^2+c^2)*a^6-2*(7*b^4-3*b^2*c^2+7*c^4)*a^4+(b^2+c^2)*(11*b^4-24*b^2*c^2+11*c^4)*a^2+2*(b^4-c^4)^2 : :

This conic is the bicevian conic of X(4) and X(598).

X(14763) lies on these lines: {4,575}, {597,8262}, {6329,8705}

X(14764) = CENTER OF THE {LEMOINE, STEINER}-CIRCUMCONIC

Barycentrics    4*a^8+7*(b^2+c^2)*a^6+2*(8*b^4-31*b^2*c^2+8*c^4)*a^4-(3*b^2-2*c^2)*(2*b^2-3*c^2)*(b^2+c^2)*a^2-(b^4+c^4)*(b^2+3*b*c+c^2)*(b^2-3*b*c+c^2) : :

This conic is the bicevian conic of X(99) and X(598).

X(14764) lies on the line {6,8591}

X(14765) = CENTER OF THE {LEMOINE, SYMMEDIAL}-CIRCUMCONIC

Barycentrics    a^2*(2*(3*b^4+2*b^2*c^2+3*c^4)*a^6-(b^2+c^2)*(9*b^4+26*b^2*c^2+9*c^4)*a^4+(3*b^8+3*c^8-10*b^2*c^2*(b^4+5*b^2*c^2+c^4))*a^2+b^2*c^2*(b^2+c^2)*(5*b^4-14*b^2*c^2+5*c^4)) : :

This conic is the bicevian conic of X(6) and X(598).

X(14765) lies on these lines: {}

X(14766) = CENTER OF THE {LEMOINE, YFF CONTACT}-CIRCUMCONIC

Barycentrics    8*a^6-8*(b+c)*a^5+4*(11*b^2-9*b*c+11*c^2)*a^4+(b+c)*(5*b^2-54*b*c+5*c^2)*a^3-(2*b^4+2*c^4-b*c*(9*b^2+50*b*c+9*c^2))*a^2-5*(b+c)*(b^2+c^2)^2*a-2*b^6+9*b^5*c+9*b*c^5-2*c^6+3*b^4*c^2+3*b^2*c^4 : :

This conic is the bicevian conic of X(190) and X(598).

X(14766) lies on these lines: {}

X(14767) = CENTER OF THE {MACBEATH, MEDIAL}-CIRCUMCONIC

Barycentrics    (b^2+c^2)*a^6-2*(b^4+b^2*c^2+c^4)*a^4+(b^4-c^4)*(b^2-c^2)*a^2+2*(b^2-c^2)^2*b^2*c^2 : :
Barycentrics    2 csc 2A + csc 2B + csc 2C : :
Barycentrics    (sin B)(sin 2B) cos(C - A) + (sin C)(sin 2C) cos(A - B) : :
X(14767) = 3*X(2)+X(264) = 9*X(2)-X(3164) = 3*X(216)-X(3164) = 3*X(264)+X(3164)

This conic is the bicevian conic of X(2) and X(264). Let P be a point on the circumcircle. The conic is also the locus of the polar conjugate of the X(4)-cross conjugate of P as P varies. (Randy Hutson, November 2, 2017)

X(14767) lies on these lines: {2,216}, {5,141}, {95,401}, {140,6709}, {183,10314}, {323,4993}, {338,570}, {458,577}, {1656,14059}, {3619,8797}, {3628,6663}, {3734,9818}, {3819,10184}, {5158,9308}, {6642,7815}, {7401,7800}, {7404,7795}, {7526,7816}

X(14767) = midpoint of X(216) and X(264)
X(14767) = reflection of X(10003) in X(3628)
X(14767) = complement of X(216)
X(14767) = centroid of {A,B,C,X(264)}
X(14767) = {X(2), X(264)}-harmonic conjugate of X(216)

X(14768) = CENTER OF THE {MACBEATH, ORTHIC}-CIRCUMCONIC

Barycentrics    (a^2+b^2-c^2)*(a^2-b^2+c^2)*((b^2+c^2)*a^6-(b^4+c^4)*a^4-2*b^2*c^2*(b^2+c^2)*a^2+(b^2-c^2)^2*b^2*c^2) : :

This conic is the bicevian conic of X(4) and X(264).

X(14768) lies on these lines: {5,14715}, {53,232}, {324,458}, {460,2387}, {2501,5943}, {9475,11197}

X(14769) = CENTER OF THE {MACBEATH, STEINER}-CIRCUMCONIC

Barycentrics    (b^2+c^2)*a^10-(3*b^4+4*b^2*c^2+3*c^4)*a^8+(b^2+c^2)*(2*b^2-3*b*c+2*c^2)*(2*b^2+3*b*c+2*c^2)*a^6-2*(b^4+b^2*c^2+c^4)*(2*b^4-3*b^2*c^2+2*c^4)*a^4+3*(b^6+c^6)*(b^2-c^2)^2*a^2-(b^4+c^4)*(b^2-c^2)^4 : :

This conic is the bicevian conic of X(99) and X(264).

X(14769) lies on these lines: {5,49}, {114,137}, {128,136}, {427,930}, {858,13372}, {5169,11671}

X(14769) = nine-point circle-inverse-of-X(110)

X(14770) = CENTER OF THE {MACBEATH, SYMMEDIAL}-CIRCUMCONIC

Barycentrics    a^2*((b^2+c^2)^3*a^8-(b^4+c^4)*(b^4+6*b^2*c^2+c^4)*a^6-(b^2+c^2)*(b^8+c^8-b^2*c^2*(5*b^4-12*b^2*c^2+5*c^4))*a^4+(b^2-c^2)^2*(b^8+c^8+2*b^2*c^2*(b^2+c^2)^2)*a^2-(-c^4+b^4)*(b^2-c^2)*b^2*c^2*(b^4-4*b^2*c^2+c^4)) : :

This conic is the bicevian conic of X(6) and X(264).

X(14770) lies on these lines: {}

X(14771) = CENTER OF THE {MACBEATH, YFF CONTACT}-CIRCUMCONIC

Barycentrics    2*(b^2+c^2)*a^8-(b+c)^3*a^7-(6*b^4+6*c^4-b*c*(3*b^2-2*b*c+3*c^2))*a^6+(b+c)*(5*b^4+2*b^2*c^2+5*c^4)*a^5+(4*b^4+4*c^4-b*c*(13*b^2-15*b*c+13*c^2))*(b+c)^2*a^4-(b^2-c^2)*(b-c)*(7*b^4+7*c^4+2*b*c*(4*b^2+3*b*c+4*c^2))*a^3+(-c^4+b^4)*(b^2-c^2)*(2*b^2+b*c+2*c^2)*a^2+(b^2-c^2)^3*(b-c)*(3*b^2+2*b*c+3*c^2)*a+(b-c)^2*(b^2-c^2)^2*(-2*b^4-2*c^4-b*c*(3*b^2+b*c+3*c^2)) : :

This conic is the bicevian conic of X(190) and X(264).

X(14771) lies on these lines: {}

X(14772) = CENTER OF THE {ORTHIC, STEINER}-CIRCUMCONIC

Barycentrics    (b^2+c^2)*a^6+(b^2-3*c^2)*(3*b^2-c^2)*a^4-(2*b^2-c^2)*(b^2-2*c^2)*(b^2+c^2)*a^2+(b^2-c^2)^2*b^2*c^2 : :

This conic is the bicevian conic of X(4) and X(99).

X(14772) lies on these lines: {2,14726}, {6,1632}, {194,1992}, {648,12829}, {670,9766}, {804,5186}, {1084,1196}

X(14772) = anticomplement of X(14726)

X(14773) = CENTER OF THE {ORTHIC, SYMMEDIAL}-CIRCUMCONIC

Barycentrics    a^2*((b^2+c^2)^2*a^6-2*(b^4+b^2*c^2+c^4)*(b^2+c^2)*a^4+(b^2-c^2)^2*(b^4+3*b^2*c^2+c^4)*a^2+(-c^4+b^4)*(b^2-c^2)*b^2*c^2) : :

This conic is the bicevian conic of X(4) and X(6).

X(14773) lies on these lines: {39,51}, {232,428}, {262,1180}, {1194,2023}, {9419,11205}

X(14774) = CENTER OF THE {ORTHIC, YFF CONTACT}-CIRCUMCONIC

Barycentrics    (5*b^2-6*b*c+5*c^2)*a^4+(b-3*c)*(3*b-c)*(b+c)*a^3-(3*b^4+3*c^4-b*c*(3*b^2+4*b*c+3*c^2))*a^2-(b^4-c^4)*(b-c)*a+(b^2-c^2)^2*b*c : :

This conic is the bicevian conic of X(4) and X(190).

X(14774) lies on the line {614,8054}

X(14775) = CENTER OF THE {PELLETIER, SCHROETER}-CIRCUMCONIC

Barycentrics    (b-c)*f(b,c,a)*f(c,a,b) : : , where f(a,b,c)=((b+c)*a^2+2*b*c*a-(b-c)*(b^2-c^2))*(-a^2+b^2+c^2)

This conic is the bianticevian conic of X(523) and X(650).

X(14775) lies on these lines: {523,2074}, {657,3064}, {663,7649}, {850,7253}, {3900,4036}

X(14775) = trilinear pole of the line {115, 5521}

X(14776) = CENTER OF THE {PELLETIER, TANGENTIAL}-CIRCUMCONIC

Barycentrics    a^2*f(b,c,a)*f(c,a,b) : : , where f(a,b,c)=(b-c)*(-a^2+b^2+c^2)*((b+c)*a^2-2*b*c*a-(b-c)*(b^2-c^2))

This conic is the bianticevian conic of X(6) and X(650).

X(14776) lies on these lines: {108,513}, {110,1309}, {663,8750}, {692,3900}, {1618,7012}, {1795,3220}, {2182,10535}

X(14776) = trilinear pole of the line {32, 607}

X(14777) = CENTER OF THE {PELLETIER, X-PARABOLA-TANGENTIAL}-CIRCUMCONIC

Barycentrics    (b-c)*f(b,c,a)*f(c,a,b) : : , where f(a,b,c)=((b+c)*a^4+2*b*c*a^3-(b+c)*(2*b^2-3*b*c+2*c^2)*a^2-b*c*(b^2+c^2)*a+(b^4-c^4)*(b-c))*(a^4-(2*b^2+b*c+2*c^2)*a^2-b*c*(b+c)*a+c^4+b^4)

This conic is the bianticevian conic of X(115) and X(650).

X(14777) lies on these lines: {}

X(14778) = CENTER OF THE {STEINER, SYMMEDIAL}-CIRCUMCONIC

Barycentrics    a^2*(b^2+c^2)*((b^2+c^2)*a^6+(b^4-4*b^2*c^2+c^4)*a^4-b^2*c^2*(b^2+c^2)*a^2+2*b^4*c^4) : :
X(14778) = X(3124)-3*X(11205)

This conic is the bicevian conic of X(6) and X(99).

X(14778) lies on these lines: {6,10330}, {732,3231}, {1194,3124}, {3051,4576}

X(14779) = CENTER OF THE {STEINER, YFF CONTACT}-CIRCUMCONIC

Barycentrics    (b-c)*(a^2+7*(b+c)*a+9*b*c+4*c^2+4*b^2) : :

This conic is the bicevian conic of X(99) and X(190).

X(14779) lies on these lines: {}

X(14779) = reflection of X(4608) in X(4988)
X(14779) = anticomplement of X(4608)
X(14779) = anticomplementary conjugate of anticomplement of X(35327)

X(14780) = CENTER OF THE {SYMMEDIAL, YFF CONTACT}-CIRCUMCONIC

Barycentrics    a^2*((b^2+c^2)^2*a^4+(b+c)*(b^2+c^2)*(b^2-4*b*c+c^2)*a^3-(b^4+c^4-b*c*(7*b^2-4*b*c+7*c^2))*b*c*a^2-b^2*c^2*(b+c)*(3*b^2-2*b*c+3*c^2)*a+b^3*c^3*(3*b^2-2*b*c+3*c^2)) : :

This conic is the bicevian conic of X(6) and X(190).

X(14780) lies on these lines: {}

X(14781) = CENTER OF THE {TANGENTIAL, X-PARABOLA-TANGENTIAL}-CIRCUMCONIC

Barycentrics    f(b,c,a)*f(c,a,b) : : , where f(a,b,c)=((-a^2+b^2+c^2)^2-b^2*c^2)*((b^2+c^2)*a^4-2*(b^4+c^4)*a^2+b^6+c^6)*(b^2-c^2)

This conic is the bianticevian conic of X(6) and X(115).

X(14781) lies on these lines: {94,1177}, {1576,8029}

leftri

Involution and related centers: X(14782)-X(14806)

rightri

This preamble and centers X(14782)-X(14806) were contributed by César Eliud Lozada, October 10, 2017.

"When several pairs of points A, A'; B, B' ; C, C′ ; . . . lying on a straight line are such that their distances from a fixed point O are connected by the relations

OA .OA' = OB.OB' = OC.OC′ = . . . ,

the points are called a range in involution." (Lachlan, R.: An Elementary Treatise on Modern Pure Geometry. MacMillan & Co., 1893, chapter 5, pp. 37-41).

A range in involution is denoted by {A, A′}, {B, B′}, {C, C′}. The point O, collinear with the given points, is called the center of involution and each pair of corresponding points, as A and A′, are said to be a conjugate-couple.

To construct O, let {A, A'}, {B, B'} be two pairs of points on a straight line. Through A and B draw any two lines AP, BP intersecting in P and through A', B' draw A'Q, B'Q parallel to BP, AP respectively, meeting in Q. PQ meets AB in O. (Proof in the above reference). From this construction, it is clear that two pairs of points are sufficient for determining the range unambigously and its notation can be shortened to {A, A'}, {B, B'}.

Some properties:

 1) Suppose A, A′, B, B′ are four distinct collinear points. Let ΓA be any circle through A and A′ and ΓB any circle through B and B′. Then, for all ΓA and ΓB, their radical axis passes through O, the center of involution of {A, A′}, {B, B′}.

 2) If all the points in a range in involution lie on the same side of the center of involution, there exist two real points, one on either side of the center, each of which coincides with its own conjugate. These points are called the double points of the involution. (Construction: from the center of involution O draw the tangent OT to any circle through a pair of conjugate points, for example A, A′. The circle (O,T) cuts AA′ in S and S′ which are the double points of the involution).

 3) Any pair of conjugate points of a range in involution are harmonic conjugates with respect to the double points of the involution.

Trilinear coordinates:

Supose Y = Uy : Vy : Wy and Z = Uz : Vz : Wz (trilinears) and let′s assume that Y′, Z′ are such that YY' = λ YZ and ZZ' = μ YZ. Let X be another point such that YX = ϰ YZ.

The center of involution of {Y, Y'}, {Z, Z'} has coordinates:
  O = Uy (λ - 1) (a Uz + b Vz + c Wz) - Uz μ (a Uy + b Vy + c Wy) : :
      = (λ - 1) Y - μ Z

If O = Uo : Vo : Wo then X', the conjugate-couple of X, has coordinates:
  X' = μ ((a Uz + b Vz + c Wz) Uy - ((Uy Vz - Uz Vy) b + (Uy Wz - Uz Wy) c) λ) + ϰ Uo : :

The double points of the involution have coordinates:
  S± = ± η ((Uy Vz - Uz Vy) b + (Uy Wz - Uz Wy) c) μ + Uz μ (a Uy + b Vy + c Wy) - Uy (λ - 1) (a Uz + b Vz + c Wz) : :
  where η = sqrt((λ - μ) (λ - 1)/μ).

The following table contains the ETC indexes of points forming some ranges in involution, their centers and double points, when these are real.

Row Range in involution Center Double points
1 {2, 3}, {4, 5}, {20, 1656}, {21, 6863}, {22, 14786}, {23, 14787}, {25, 7404}, {30, 3090}, {140, 631}, {376, 3628}, {377, 6882}, {381, 3091}, {382, 5056}, {404, 6958}, {405, 6825}, {411, 6861}, {427, 7401}, {442, 6827}, {443, 6922}, {474, 6891}, {546, 3545}, {547, 3529}, {549, 3525}, {550, 5067}, {632, 3524}, {1012, 6944}, {1368, 6803}, {1370, 7405}, {1532, 6893}, {1595, 7392}, {1657, 7486}, {2041, 2046}, {2042, 2045}, {2475, 6971}, {2476, 6928}, {2478, 6842}, {3088, 5020}, {3089, 11479}, {3146, 5055}, {3149, 6824}, {3522, 5070}, {3523, 3526}, {3530, 3533}, {3541, 6642}, {3542, 9818}, {3543, 5079}, {3544, 3845}, {3547, 7395}, {3549, 7503}, {3560, 6834}, {3627, 5071}, {3832, 3851}, {3839, 5072}, {3843, 5068}, {3850, 3855}, {4187, 6850}, {4193, 6923}, {5046, 6980}, {5054, 10303}, {5084, 6907}, {5133, 7528}, {5576, 7544}, {6643, 7399}, {6675, 6988}, {6804, 6823}, {6815, 11585}, {6826, 6831}, {6830, 6917}, {6832, 6985}, {6833, 6911}, {6835, 6841}, {6836, 6881}, {6847, 6918}, {6848, 6913}, {6849, 8226}, {6862, 6905}, {6864, 8727}, {6865, 8728}, {6883, 6889}, {6887, 7580}, {6906, 6959}, {6908, 11108}, {6914, 6949}, {6924, 6952}, {6929, 6941}, {6933, 7491}, {6954, 7483}, {6960, 7489}, {6961, 13747}, {6978, 11112}, {6979, 13743}, {6997, 7403}, {7383, 7393}, {7505, 7526}, {7514, 7558}, {8889, 9825} 3090 14782, 14783
2 {2, 4}, {3, 5}, {20, 3090}, {21, 6830}, {22, 14788}, {23, 14789}, {24, 13160}, {25, 7399}, {30, 1656}, {140, 381}, {376, 5056}, {377, 6834}, {382, 3628}, {383, 11289}, {404, 6941}, {405, 6831}, {411, 6829}, {427, 7395}, {442, 3149}, {443, 6848}, {452, 6956}, {474, 1532}, {546, 3526}, {547, 1657}, {548, 5079}, {549, 3851}, {550, 5055}, {631, 3091}, {632, 3843}, {1006, 6828}, {1012, 4187}, {1080, 11290}, {1368, 11479}, {1513, 7770}, {1585, 6810}, {1586, 6809}, {1594, 7503}, {2043, 2046}, {2044, 2045}, {2050, 4205}, {2072, 7526}, {2475, 6949}, {2476, 6905}, {2478, 6833}, {3088, 6804}, {3089, 6803}, {3146, 5067}, {3522, 5071}, {3523, 3545}, {3524, 5068}, {3525, 3832}, {3529, 7486}, {3530, 5072}, {3533, 3839}, {3541, 6816}, {3542, 6815}, {3547, 7401}, {3560, 6882}, {3627, 5070}, {3651, 6991}, {3850, 5054}, {3855, 10303}, {4193, 6906}, {5020, 6823}, {5046, 6952}, {5047, 6845}, {5084, 6847}, {5125, 7567}, {5133, 7509}, {5141, 6942}, {5142, 7549}, {5154, 6950}, {5169, 7550}, {5177, 6927}, {5187, 6977}, {5576, 7514}, {6643, 7404}, {6644, 10024}, {6656, 13860}, {6805, 6808}, {6806, 6807}, {6811, 7388}, {6813, 7389}, {6824, 6827}, {6825, 6826}, {6832, 6836}, {6835, 6889}, {6837, 6947}, {6838, 6854}, {6839, 6853}, {6840, 6852}, {6841, 6883}, {6842, 6911}, {6843, 6988}, {6844, 6857}, {6846, 6865}, {6849, 6989}, {6850, 6944}, {6851, 6887}, {6855, 6987}, {6858, 6869}, {6859, 6868}, {6860, 6936}, {6862, 6928}, {6863, 6917}, {6864, 6908}, {6867, 6954}, {6870, 6878}, {6871, 6880}, {6872, 6879}, {6881, 6985}, {6884, 6903}, {6886, 6899}, {6888, 6902}, {6890, 6898}, {6891, 6893}, {6897, 6953}, {6901, 6960}, {6904, 6969}, {6907, 6918}, {6909, 6975}, {6912, 6963}, {6913, 6922}, {6914, 6971}, {6915, 6937}, {6916, 6964}, {6919, 6935}, {6920, 6943}, {6921, 6968}, {6923, 6959}, {6924, 6980}, {6925, 6983}, {6926, 6939}, {6929, 6958}, {6930, 6978}, {6931, 6938}, {6932, 6946}, {6933, 6934}, {6940, 6945}, {6948, 6981}, {6951, 6979}, {6957, 6967}, {6961, 6973}, {6962, 6984}, {6965, 6972}, {6970, 6982}, {6986, 6990}, {6996, 7380}, {6997, 7383}, {6998, 7377}, {7000, 7375}, {7374, 7376}, {7387, 7405}, {7390, 7402}, {7392, 7400}, {7393, 7403}, {7397, 7407}, {7544, 7558}, {7569, 12225}, {8727, 11108}, {9818, 11585}, {14782, 14783} 1656 Imaginary
3 {2, 5}, {3, 4}, {20, 30}, {21, 6917}, {22, 14790}, {23, 14791}, {25, 6643}, {140, 3091}, {235, 3546}, {376, 382}, {377, 3560}, {381, 631}, {403, 3548}, {404, 6929}, {405, 6826}, {427, 3547}, {442, 6824}, {443, 6913}, {464, 7534}, {474, 6893}, {546, 3523}, {547, 7486}, {548, 3543}, {549, 3832}, {550, 3146}, {632, 5068}, {1012, 6850}, {1368, 3089}, {1370, 7387}, {1532, 6891}, {1594, 3549}, {1595, 7400}, {1597, 10996}, {1598, 7386}, {1656, 3090}, {1657, 3529}, {1658, 3153}, {2041, 2043}, {2042, 2044}, {2072, 7505}, {2475, 6914}, {2476, 6862}, {2478, 6911}, {3088, 6823}, {3149, 6827}, {3522, 3627}, {3524, 3843}, {3525, 3851}, {3526, 3545}, {3528, 3830}, {3530, 3839}, {3533, 5072}, {3542, 11585}, {3628, 5056}, {3850, 10303}, {3853, 10304}, {3855, 5054}, {4187, 6944}, {4193, 6959}, {5020, 6804}, {5046, 6924}, {5055, 5067}, {5070, 5071}, {5084, 6918}, {5576, 7558}, {6143, 10254}, {6639, 7577}, {6642, 6816}, {6675, 6843}, {6803, 11479}, {6815, 9818}, {6825, 6831}, {6829, 6861}, {6830, 6863}, {6832, 6881}, {6833, 6842}, {6834, 6882}, {6835, 6883}, {6836, 6985}, {6841, 6889}, {6846, 8728}, {6847, 6907}, {6848, 6922}, {6851, 7580}, {6864, 11108}, {6867, 7483}, {6885, 11113}, {6901, 7489}, {6905, 6928}, {6906, 6923}, {6908, 8727}, {6930, 11112}, {6934, 7491}, {6941, 6958}, {6949, 6971}, {6951, 13743}, {6952, 6980}, {6973, 13747}, {6989, 8226}, {6997, 7393}, {7383, 7403}, {7394, 7516}, {7395, 7401}, {7399, 7404}, {7487, 12362}, {7509, 7528}, {7514, 7544}, {7556, 7574}, {8613, 11348}, {14782, 14783} 20 14784, 14785
4 {1, 3}, {35, 36}, {40, 14793}, {55, 7280}, {56, 5010}, {65, 14794}, {165, 8071}, {517, 14792}, {7987, 8069} 14792 Imaginary
5 {1, 35}, {3, 36}, {40, 14798}, {55, 3746}, {65, 14799}, {517, 14795}, {5010, 5563}, {5903, 10902} 14795 14796, 14797
6 {1, 36}, {3, 35}, {40, 14803}, {56, 5563}, {65, 14804}, {517, 14800}, {3746, 7280}, {14792, 14795}, {14793, 14798}, {14794, 14799}, {14796, 14797} 14800 14801, 14802
7 {3, 6}, {15, 16}, {32, 5092}, {39, 3098}, {50, 14805}, {52, 14806}, {61, 10646}, {62, 10645}, {182, 187}, {216, 11438}, {371, 6396}, {372, 6200}, {389, 10979}, {511, 574}, {566, 3581}, {572, 4257}, {573, 4256}, {575, 8588}, {576, 8589}, {577, 11430}, {991, 5030}, {1151, 6398}, {1152, 6221}, {1340, 1379}, {1341, 1380}, {1350, 5024}, {1384, 5085}, {3053, 12017}, {3311, 6412}, {3312, 6411}, {3592, 6452}, {3594, 6451}, {4262, 13329}, {5033, 13335}, {5050, 5210}, {5063, 10564}, {5104, 11171}, {5107, 9734}, {5116, 9301}, {6199, 6410}, {6395, 6409}, {6407, 6469}, {6408, 6468}, {6425, 6446}, {6426, 6445}, {6430, 9690}, {6437, 6450}, {6438, 6449}, {6440, 9691}, {6453, 6481}, {6454, 6480}, {9600, 9733}, {9821, 12055}, {11480, 11486}, {11481, 11485} 574 Imaginary

X(14782) = 1st DOUBLE POINT OF THE RANGE IN INVOLUTION {X(2), X(3)}, {X(4), X(5)}

Trilinears         cos(A)+(sqrt(2)-1)*cos(B-C) : :
Barycentrics    (sqrt(2)-1)*SB*SC-S^2 : :
X(14782) = (2+sqrt(2))*X(3)+X(4)

As a point on the Euler line, X(14782) has Shinagawa coefficients (sqrt(2)+4, -3*sqrt(2)+2)

X(14782) lies on these lines: {2,3}, {485,3372}, {486,3385}, {2575,5732}, {3371,5420}, {3374,6561}, {3386,5418}, {3387,6560}, {7741,14802}, {7951,14797}

X(14782) = reflection of X(14783) in X(3090)
X(14782) = orthocentroidal circle-inverse-of-X(14784)
X(14782) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (382, 11098, 5154), (452, 6893, 7866), (1532, 10301, 3861), (1557, 13632, 440), (1594, 7461, 6982), (1650, 6829, 1599), (3138, 6642, 7841), (6991, 20101, 447), (7535, 7555, 12105)
X(14782) = {X(i),X(j)}-harmonic conjugate of X(14783) for all {i, j} in the range in involution listed in row 1 of the table showed in the preamble just before X(14782)

X(14783) = 2nd DOUBLE POINT OF THE RANGE IN INVOLUTION {X(2), X(3)}, {X(4), X(5)}

Trilinears         cos(A)-(sqrt(2)+1)*cos(B-C) : :
Barycentrics    (sqrt(2)+1)*SB*SC+S^2 : :
X(14783) = (2-sqrt(2))*X(3)+X(4)

As a point on the Euler line, X(14783) has Shinagawa coefficients (-sqrt(2)+4, 3*sqrt(2)+2)

X(14783) lies on these lines: {2,3}, {485,3371}, {486,3386}, {3373,6561}, {3385,5418}, {3388,6560}, {7741,14801}, {7951,14796}, {10194,12823}, {10195,12822}

X(14783) = reflection of X(14782) in X(3090)
X(14783) = orthocentroidal circle-inverse-of-X(14785)
X(14783) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6888, 8366, 4242), (7424, 11326, 410), (7478, 11313, 7460), (7576, 13860, 13723), (11100, 12068, 14039)
X(14783) = {X(i),X(j)}-harmonic conjugate of X(14782) for all {i, j} in the range in involution listed in row 1 of the table showed in the preamble just before X(14782)

X(14784) = 1st DOUBLE POINT OF THE RANGE IN INVOLUTION {X(2), X(5)}, {X(3), X(4)}

Trilinears         cos(B-C)+sqrt(2)*sin(B)*sin(C) : :
Barycentrics    SB*SC+(sqrt(2)+1)*S^2 : :
X(14784) = sqrt(2)*X(3)+X(4)

As a point on the Euler line, X(14784) has Shinagawa coefficients (2-sqrt(2), -4+3*sqrt(2))

X(14784) lies on the cubic K690 and these lines: {2,3}, {485,3386}, {486,3371}, {3372,6560}, {3374,5418}, {3385,6561}, {3387,5420}, {7741,14796}, {7951,14801}, {10483,14797}

X(14784) = reflection of X(14785) in X(20)
X(14784) = orthocentroidal circle-inverse-of-X(14782)
X(14784) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (5, 7491, 7545), (414, 11832, 853), (2675, 3136, 12056), (5142, 6975, 8964), (5177, 6850, 7819), (6916, 11319, 8360)
X(14784) = {X(i),X(j)}-harmonic conjugate of X(14785) for all {i, j} in the range in involution listed in row 3 of the table showed in the preamble just before X(14782)

X(14785) = 2nd DOUBLE POINT OF THE RANGE IN INVOLUTION {X(2), X(5)}, {X(3), X(4)}

Trilinears         cos(B-C)-sqrt(2)*sin(B)*sin(C) : :
Barycentrics    SB*SC-(sqrt(2)-1)*S^2 : :
X(14785) = sqrt(2)*X(3)-X(4)

As a point on the Euler line, X(14785) has Shinagawa coefficients (2+sqrt(2), -4-3*sqrt(2))

X(14785) lies on the cubic K690 and these lines: {2,3}, {485,3385}, {3371,6560}, {3373,5418}, {3386,6561}, {3388,5420}, {7741,14797}, {7951,14802}, {10194,12822}, {10195,12823}, {10483,14796}

X(14785) = reflection of X(14784) in X(20)
X(14785) = orthocentroidal circle-inverse-of-X(14783)
X(14785) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2454, 6826, 14009), (6636, 6911, 10126), (7396, 13587, 13154), (7497, 10254, 407), (7542, 10128, 10221), (11114, 11146, 7524)
X(14785) = {X(i),X(j)}-harmonic conjugate of X(14784) for all {i, j} in the range in involution listed in row 3 of the table showed in the preamble just before X(14782)

X(14786) = CONJUGATE-COUPLE OF X(22) IN THE RANGE IN INVOLUTION {X(2), X(3)}, {X(4), X(5)}

Trilinears         8*cos(B-C)+cos(A)*(2*cos(2*(B-C))+3)-cos(3*A) : :
Barycentrics    R^2*SB*SC-(R^2-SW)*S^2 : :
X(14786) = 2*(R^2-SW)*X(3)-SW*X(4)

As a point on the Euler line, X(14786) has Shinagawa coefficients (3*E+4*F, E)

X(14786) lies on these lines: {2,3}, {32,233}, {39,2165}, {206,13336}, {567,6193}, {569,1352}, {3618,11411}, {5050,13562}, {5266,10320}, {6776,13353}, {9936,13366}, {9969,10625}, {10516,12134}, {10601,12359}, {11427,11487}, {14376,14767}

X(14786) = complement of X(7383)
X(14786) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 7404, 3), (3, 381, 7553), (4, 631, 6636), (5, 26, 6997), (5, 6676, 7529), (30, 7431, 10109), (140, 1595, 3), (3525, 7493, 7568), (3628, 3843, 186), (5133, 7509, 14790), (6839, 6929, 2475), (7477, 12083, 3542), (14782, 14783, 22)

X(14787) = CONJUGATE-COUPLE OF X(23) IN THE RANGE IN INVOLUTION {X(2), X(3)}, {X(4), X(5)}

Trilinears         8*cos(B-C)+cos(A)*(2*cos(2*(B-C))+1)-cos(3*A) : :
Barycentrics    3*R^2*SB*SC-(3*R^2-2*SW)*S^2 : :
X(14787) = (3*R^2-2*SW)*X(3)-SW*X(4) = X(3)+2*X(7403) = 2*X(5)+X(7503)

As a point on the Euler line, X(14787) has Shinagawa coefficients (5*E+8*F, 3*E)

X(14787) lies on these lines: {2,3}, {52,5476}, {542,569}, {567,1352}, {574,1879}, {578,11178}, {10168,13336}, {11179,13353}

X(14787) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 3091, 7552), (2, 3545, 10201), (3, 381, 7540), (4, 631, 7492), (381, 5054, 9909), (1656, 11479, 10024), (1995, 7503, 10323), (2043, 2044, 11818), (5169, 7550, 14791), (7404, 14786, 3), (7527, 7570, 14789), (13615, 14066, 6945), (14782, 14783, 23)

X(14788) = CONJUGATE-COUPLE OF X(22) IN THE RANGE IN INVOLUTION {X(2), X(4)}, {X(3), X(5)}

Trilinears         (cos(2*A)-2)*cos(B-C)-cos(A)*(cos(2*(B-C))+2) : :
Barycentrics    (2*R^2-SW)*SB*SC-S^2*SW : :
X(14788) = SW*X(3)-(R^2-SW)*X(4)

As a point on the Euler line, X(14788) has Shinagawa coefficients (2*E+2*F, E+2*F)

X(14788) lies on these lines: {2,3}, {68,5422}, {76,1238}, {83,96}, {125,11695}, {141,11412}, {233,5523}, {343,3567}, {569,14516}, {1147,14389}, {1181,10516}, {1199,3564}, {1209,3580}, {1352,7592}, {2888,13292}, {2965,7745}, {3292,12242}, {3574,11793}, {3589,6146}, {3818,10984}, {5012,12134}, {5254,13351}, {5286,9722}, {8068,13161}, {11459,12233}, {11804,13392}

X(14788) = complement of X(37126)
X(14788) = orthocentroidal circle-inverse-of-X(7509)
X(14788) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 4, 7509), (4, 7383, 10323), (4, 7400, 12082), (4, 7550, 12362), (4, 14789, 7399), (5, 235, 3545), (5, 11563, 12811), (140, 5576, 858), (140, 11819, 3), (2567, 6852, 7501), (3547, 6997, 10594), (3832, 6906, 3858), (7485, 7566, 14790)

X(14789) = CONJUGATE-COUPLE OF X(23) IN THE RANGE IN INVOLUTION {X(2), X(4)}, {X(3), X(5)}

Trilinears         (2*cos(2*A)-3)*cos(B-C)-cos(A)*(2*cos(2*(B-C))+5) : :
Barycentrics    (3*R^2-SW)*SB*SC-S^2*SW : :
X(14789) = 2*SW*X(3)-(3*R^2-2*SW)*X(4)

As a point on the Euler line, X(14789) has Shinagawa coefficients (4*E+4*F, E+4*F)

X(14789) lies on these lines: {2,3}, {76,1273}, {3574,7999}, {3589,12022}, {6689,11449}, {10516,11456}

X(14789) = orthocentroidal circle-inverse-of-X(7550)
X(14789) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 4, 7550), (2, 11347, 10018), (5, 11799, 3091), (381, 12082, 4), (472, 4239, 7380), (1559, 11548, 6817), (1567, 4201, 2), (3853, 11548, 7428), (4222, 10204, 11105), (4244, 11286, 6871), (7496, 7565, 14791), (7527, 7570, 14787), (14782, 14783, 14791)

X(14790) = CONJUGATE-COUPLE OF X(22) IN THE RANGE IN INVOLUTION {X(2), X(5)}, {X(3), X(4)}

Trilinears         4*cos(2*A)*cos(B-C)-cos(A)*(2*cos(2*(B-C))-3)-cos(3*A) : :
Barycentrics    (R^2-SW)*SB*SC+R^2*S^2 : :
X(14790) = 9*X(2)-8*X(10020) = 3*X(2)-4*X(13371) = 2*R^2*X(3)+(2*R^2-SW)*X(4) = 3*X(154)-4*X(9820) = 4*X(156)-3*X(11206) = 3*X(1853)-2*X(12359) = 3*X(1992)-4*X(11255) = 3*X(5654)-2*X(6759)

As a point on the Euler line, X(14790) has Shinagawa coefficients (-E, 3*E+4*F)

X(14790) lies on these lines: {2,3}, {11,9645}, {52,1899}, {66,68}, {69,6101}, {143,11433}, {154,9820}, {155,1503}, {156,11206}, {394,12134}, {497,8144}, {542,9936}, {570,2548}, {571,3767}, {1060,11392}, {1062,11393}, {1147,9833}, {1154,11411}, {1216,1352}, {1351,13292}, {1853,12359}, {1992,11255}, {2322,3308}, {2550,8141}, {3069,11266}, {3564,12318}, {3574,10984}, {3818,11793}, {5562,11550}, {5654,6759}, {5663,12319}, {5889,11457}, {5892,9815}, {6243,6515}, {6247,12163}, {6288,13340}, {6776,12161}, {9306,13419}, {11412,11442}, {11750,13352}, {12118,13346}, {13754,14216}

X(14790) = midpoint of X(12319) and X(13203)
X(14790) = reflection of X(i) in X(j) for these (i,j): (20, 12084), (7387, 5), (9833, 1147), (12118, 13346), (12163, 6247)
X(14790) = isogonal conjugate of X(34439)
X(14790) = complement of X(31305)
X(14790) = anticomplement of X(26)
X(14790) = anticomplementary circle-inverse-of-X(186)
X(14790) = orthocentroidal circle-inverse-of-X(7528)
X(14790) = X(5)-of-polar-triangle-of-anticomplementary-circle
X(14790) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2,4,7528), (3,4,18420), (3, 382, 3575), (3, 5576, 2), (22, 1594, 3549), (382, 7506, 7540), (631, 7526, 5068), (1595, 12362, 9818), (2937, 6639, 7493), (3146, 3153, 4), (3530, 5073, 7488), (3542, 7500, 7517), (5064, 7395, 7403), (7553, 11585, 25), (10750, 10751, 10151)

X(14791) = CONJUGATE-COUPLE OF X(23) IN THE RANGE IN INVOLUTION {X(2), X(5)}, {X(3), X(4)}

Trilinears         2*(2*cos(2*A)+1)*cos(B-C)-cos(A)*(2*cos(2*(B-C))-3)-cos(3*A) : :
Barycentrics    (3*R^2-2*SW)*SB*SC+3*R^2*S^2 : :
X(14791) = 3*R^2*X(3)+(3*R^2-SW)*X(4) = X(4)+3*X(1370) = 3*X(25)-5*X(1656)

As a point on the Euler line, X(14791) has Shinagawa coefficients (-3*E, 5*E+8*F)

X(14791) lies on these lines: {2,3}, {50,3767}, {67,68}, {70,3519}, {265,13340}, {566,2548}, {1092,11750}, {1154,1899}, {1216,2393}, {3098,6697}, {3574,13336}, {3818,10170}, {5095,8538}, {5656,13203}, {5876,14216}, {5891,11550}, {6103,10316}, {8550,12161}, {10264,12319}, {11202,14156}

X(14791) = reflection of X(i) in X(j) for these (i,j): (6644, 1368), (7530, 5)
X(14791) = anticomplement of X(12106)
X(14791) = Droz-Farny 1st circle-inverse-of-X(14120)
X(14791) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 1656, 7495), (4, 3090, 7533), (20, 7505, 2937), (22, 2072, 10201), (426, 6952, 11314), (5169, 7550, 14787), (6875, 13735, 7508), (7496, 7565, 14789), (7516, 7530, 6644), (7833, 12811, 1599), (10295, 12225, 1657), (14782, 14783, 14789), (14784, 14785, 23)

X(14792) = CENTER OF INVOLUTION OF {X(1), X(3)}, {X(35), X(36)}

Trilinears    4*cos(A)*(2*cos((B-C)/2)*sin(A/2)-1)+2*cos(2*A)+3 : :
X(14792) = R^2*X(1)+4*r^2*X(3) = (R-2*r)*R*X(1)+2*(R+2*r)*r*X(35)

X(14792) lies on these lines: {1,3}, {10,4996}, {21,3825}, {30,8070}, {140,8068}, {404,3822}, {411,4316}, {498,4188}, {499,4189}, {549,10523}, {1193,2964}, {1449,8553}, {1478,6942}, {1479,6950}, {1737,5267}, {1785,3520}, {1898,3065}, {1935,6127}, {3523,10320}, {3524,10629}, {3583,6906}, {3584,13587}, {3585,6905}, {3663,7279}, {4324,6909}, {4857,10058}, {5433,7508}, {5441,10073}, {6914,7741}, {6924,7951}, {10265,10572}

X(14792) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 14793, 1), (35, 36, 1385), (3336, 7280, 36), (5010, 7280, 7987), (14796, 14797, 14800), (14801, 14802, 14795)

X(14793) = CONJUGATE-COUPLE OF X(40) IN THE RANGE IN INVOLUTION {X(1), X(3)}, {X(35), X(36)}

Trilinears    2*cos(A)*(2*cos((B-C)/2)*sin(A/2)-1)+cos(2*A)+2 : :
X(14793) = R^2*X(1)+2*r^2*X(3)

X(14793) lies on these lines: {1,3}, {2,4996}, {4,8070}, {11,6914}, {12,6924}, {21,499}, {80,11502}, {100,12647}, {140,10523}, {378,1785}, {388,6942}, {404,498}, {411,4299}, {497,5533}, {574,13006}, {611,5096}, {613,4265}, {631,10320}, {909,4266}, {920,4652}, {993,1737}, {1012,3583}, {1100,8553}, {1210,5267}, {1376,10057}, {1449,1609}, {1478,6905}, {1479,6906}, {1599,5405}, {1600,5393}, {1790,4276}, {1795,3422}, {2932,4421}, {2975,10573}, {3085,4188}, {3086,4189}, {3149,3585}, {3520,7952}, {3523,10321}, {3560,7741}, {3672,7279}, {4302,6909}, {4316,7580}, {4357,9723}, {5450,10572}, {6875,7288}, {6911,7951}, {6958,10953}, {6980,13273}, {6985,10483}, {7509,13161}, {7967,10074}, {10073,11219}, {10896,13743}

X(14793) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 14792, 3), (3, 999, 5172), (3, 1470, 36), (3, 8071, 1), (35, 36, 3576), (36, 5902, 56), (55, 56, 10246), (56, 11507, 1), (3576, 5535, 11014), (7280, 14794, 3), (10966, 11248, 5697), (11249, 11509, 5903), (14801, 14802, 14798)

X(14794) = CONJUGATE-COUPLE OF X(65) IN THE RANGE IN INVOLUTION {X(1), X(3)}, {X(35), X(36)}

Trilinears    8*sin(A/2)*cos(A)*cos((B-C)/2)+2*cos(2*A)+3 : :
X(14794) = R^2*X(1)+4*r*(R+r)*X(3)

X(14794) lies on these lines: {1,3}, {11,5428}, {21,3583}, {60,4276}, {225,3520}, {283,501}, {411,3585}, {993,5086}, {1030,2323}, {1064,2964}, {1478,6876}, {1479,6875}, {1749,1858}, {3074,6127}, {3651,4316}, {4188,10198}, {4189,4302}, {4216,9591}, {4265,9047}, {4292,14526}, {4324,6906}, {4330,10058}, {4996,5267}, {5258,9897}, {5427,12433}, {6284,7508}, {6986,10090}

X(14794) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 14793, 7280), (35, 36, 2646), (35, 11009, 55), (35, 11012, 1), (3337, 7280, 36), (3612, 5709, 1), (5010, 11010, 35), (5217, 11849, 35), (14796, 14797, 14804), (14801, 14802, 14799)

X(14795) = CENTER OF INVOLUTION OF {X(1), X(35)}, {X(3), X(36)}

Trilinears    4*sin(3*A/2)*cos((B-C)/2)+2*cos(2*A)-1 : :
X(14795) = R*(R-2*r)*X(1)-2*r*(R+2*r)*X(3)

X(14795) lies on these lines: {1,3}, {80,11491}, {100,5445}, {498,6902}, {1479,6960}, {1484,5433}, {1621,5443}, {2594,2964}, {2975,7972}, {4193,5259}, {5046,5248}, {5303,10074}, {5399,6149}, {6796,6949}, {6875,12647}, {7508,10944}, {10197,11114}

X(14795) = midpoint of X(14796) and X(14797)
X(14795) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 5172, 14804), (3, 55, 11010), (35, 10902, 14799), (35, 14798, 36), (36, 3746, 11009), (14801, 14802, 14792)

X(14796) = 1st DOUBLE POINT OF THE RANGE IN INVOLUTION {X(1), X(35)}, {X(3), X(36)}

Trilinears    1 + sqrt(2) + 2 cos A : :
X(14796) = (R+2*r)*X(35)+(sqrt(2)-1)*(R-2*r)*X(36)

See the trilinears at X(14797), X(14801), X(14802).

X(14796) lies on these lines: {1,3}, {3299,3386}, {3301,3371}, {7741,14784}, {7951,14783}, {10483,14785}

X(14796) = reflection of X(14797) in X(14795)
X(14796) = {X(i),X(j)}-harmonic conjugate of X(14797) for all {i, j} in the range in involution listed in row 5 of the table showed in the preamble just before X(14782)

X(14797) = 2nd DOUBLE POINT OF THE RANGE IN INVOLUTION {X(1), X(35)}, {X(3), X(36)}

Trilinears    1 - sqrt(2) + 2 cos A : :
X(14797) = (R+2*r)*X(35)-(sqrt(2)+1)*(R-2*r)*X(36)

See the trilinears at X(14796), X(14801), X(14802).

X(14797) lies on these lines: {1,3}, {3299,3385}, {3301,3372}, {7741,14785}, {7951,14782}, {10483,14784}

X(14797) = reflection of X(14796) in X(14795)
X(14797) = {X(i),X(j)}-harmonic conjugate of X(14796) for all {i, j} in the range in involution listed in row 5 of the table showed in the preamble just before X(14782)

X(14798) = CONJUGATE-COUPLE OF X(40) IN THE RANGE IN INVOLUTION {X(1), X(35)}, {X(3), X(36)}

Trilinears    2*sin(3*A/2)*cos((B-C)/2)+cos(A)+cos(2*A)-1 : :
X(14798) = R*(R-r)*X(1)-2*r*(R+r)*X(3) = r*(R+2*r)*X(35)-R*(R-2*r)*X(36)

X(14798) lies on these lines: {1,3}, {80,10395}, {100,10916}, {140,10957}, {227,1718}, {405,10827}, {498,6947}, {499,6796}, {902,4303}, {1006,10039}, {1478,5248}, {1479,6838}, {1621,12047}, {1737,11491}, {1745,8616}, {2003,2964}, {2323,4268}, {2342,7727}, {2361,5399}, {2478,5259}, {3085,6992}, {3193,4278}, {3476,6875}, {3826,13747}, {4187,6668}, {4293,10587}, {4297,10058}, {4299,10532}, {5288,7972}, {5433,10943}, {5445,6734}, {6834,7741}, {6880,12116}, {6883,11501}, {6938,10483}, {7288,10806}, {8715,12649}, {10087,11362}, {10826,11500}

X(14798) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 7280, 11249), (3, 11510, 1), (36, 3746, 5903), (36, 14795, 35), (46, 7742, 36), (55, 7742, 46), (1159, 1735, 3749), (1319, 5119, 5425), (1617, 11507, 3338), (2078, 10902, 1), (5119, 11508, 3746), (5348, 9627, 3660), (14801, 14802, 14793)

X(14799) = CONJUGATE-COUPLE OF X(65) IN THE RANGE IN INVOLUTION {X(1), X(35)}, {X(3), X(36)}

Trilinears    4*sin(3*A/2)*cos((B-C)/2)+4*cos(A)+2*cos(2*A)+1 : :
X(14799) = R*(R+2*r)*X(1)+2*r*(3*R+2*r)*X(3)

X(14799) lies on these lines: {1,3}, {80,1006}, {411,5443}, {498,6903}, {500,2964}, {993,6224}, {2475,5248}, {2476,5259}, {4184,5127}, {4302,6951}, {5444,6905}, {5445,6986}, {5499,6284}, {6796,6952}, {6840,7951}, {6853,7741}

X(14799) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 55, 484), (35, 10902, 14795), (35, 14798, 3746), (36, 3746, 5425), (14796, 14797, 65), (14801, 14802, 14794)

X(14800) = CENTER OF INVOLUTION OF {X(1), X(36)}, {X(3), X(35)}

Trilinears    4*(2*sin(A/2)-sin(3*A/2))*cos((B-C)/2)+8*cos(A)-2*cos(2*A)-3 : :
X(14800) = R*(R+2*r)*X(1)+2*r*(R-2*r)*X(3)

X(14800) lies on these lines: {1,3}, {21,5444}, {80,404}, {186,1866}, {499,6951}, {603,6126}, {908,5267}, {1478,6972}, {1830,3520}, {2475,7741}, {4299,6903}, {5253,10058}, {5427,11277}, {5433,5499}, {5443,6906}, {5445,6940}, {5450,6952}, {6713,8070}, {6840,10483}, {12409,12524}

X(14800) = midpoint of X(14801) and X(14802)
X(14800) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 5010, 11849), (1, 11010, 10284), (3, 56, 484), (36, 14803, 35), (2646, 5885, 1), (14796, 14797, 14792)

X(14801) = 1st DOUBLE POINT OF THE RANGE IN INVOLUTION {X(1), X(36)}, {X(3), X(35)}

Trilinears    1 + sqrt(2) - 2 cos A : :    (Peter Moses, October 11, 2017; cf. X(14796, X(14797), X(14802))
Trilinears    16*p^3*(p-q)-k*(k+4*p*q) : : , where k=sqrt(2)-1, p=sin(A/2), q=cos((B-C)/2)
Trilinears    (a^5-(b+c)*a^4-2*(b-c)^2*a^3+(b+c)*(2*b^2-3*b*c+2*c^2)*a^2+(b^4+c^4-b*c*(4*b^2-3*b*c+4*c^2))*a-(b^3+c^3)*(b-c)^2-sqrt(2)*b*c*((b+c)*a^2-2*b*c*a-(b^2-c^2)*(b-c)))*a : :
X(14801) = (sqrt(2)-1)*(R+2*r)*X(35)+(R-2*r)*X(36)

X(14801) lies on these lines: {1,3}, {3299,3371}, {3301,3386}, {7741,14783}, {7951,14784}

X(14801) = reflection of X(14802) in X(14800)
X(14801) = {X(i),X(j)}-harmonic conjugate of X(14802) for all {i, j} in the range in involution listed in row 6 of the table showed in the preamble just before X(14782)

X(14802) = 2nd DOUBLE POINT OF THE RANGE IN INVOLUTION {X(1), X(36)}, {X(3), X(35)}

Trilinears    1 - sqrt(2) - 2 cos A : :    (Peter Moses, October 11, 2017; cf. X(14796, X(14797), X(14801))
Trilinears    16*p^3*(p-q)-k*(k-4*p*q) : : , where k=sqrt(2)+1, p=sin(A/2), q=cos((B-C)/2)
Trilinears    (a^5-(b+c)*a^4-2*(b-c)^2*a^3+(b+c)*(2*b^2-3*b*c+2*c^2)*a^2+(b^4+c^4-b*c*(4*b^2-3*b*c+4*c^2))*a-(b^3+c^3)*(b-c)^2+sqrt(2)*b*c*((b+c)*a^2-2*b*c*a-(b^2-c^2)*(b-c)))*a : :
X(14802) = (sqrt(2)+1)*(R+2*r)*X(35)-(R-2*r)*X(36)

X(14802) lies on these lines: {1,3}, {3299,3372}, {3301,3385}, {7741,14782}, {7951,14785}

X(14802) = reflection of X(14801) in X(14800)
X(14802) = {X(i),X(j)}-harmonic conjugate of X(14801) for all {i, j} in the range in involution listed in row 6 of the table showed in the preamble just before X(14782)

X(14803) = CONJUGATE-COUPLE OF X(40) IN THE RANGE IN INVOLUTION {X(1), X(36)}, {X(3), X(35)}

Trilinears    (4*sin(A/2)-2*sin(3*A/2))*cos((B-C)/2)+5*cos(A)-cos(2*A)-1 : :
X(14803) = R*(R+r)*X(1)+2*r*(R-r)*X(3) = R*(R+2*r)*X(35)+r*(R-2*r)*X(36)

X(14803) lies on these lines: {1,3}, {104,10039}, {140,10958}, {377,7741}, {404,10572}, {442,6667}, {474,10826}, {498,5450}, {499,6897}, {950,10090}, {960,1727}, {993,5552}, {1125,10058}, {1478,6890}, {1479,4190}, {1519,5443}, {1737,6940}, {1877,7414}, {2932,5836}, {2975,10915}, {3616,10940}, {4188,4305}, {4294,10586}, {4299,6899}, {4302,10531}, {5218,10805}, {5251,6910}, {5259,5444}, {5267,12527}, {5432,10942}, {6256,6833}, {6836,10483}, {6906,12608}, {6909,12047}, {8666,12648}, {10087,13607}, {10827,12114}

X(14803) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 3359, 5903), (1, 5010, 11248), (1, 7280, 1470), (3, 3612, 35), (35, 5563, 5697), (35, 14800, 36), (46, 14110, 3245), (14796, 14797, 14793), (14801, 14802, 40)

X(14804) = CONJUGATE-COUPLE OF X(65) IN THE RANGE IN INVOLUTION {X(1), X(36)}, {X(3), X(35)}

Trilinears    (8*sin(A/2)-4*sin(3*A/2))*cos((B-C)/2)+4*cos(A)-2*cos(2*A)-1 : :
X(14804) = R*(3*R+2*r)*X(1)-2*r*(R+2*r)*X(3) = R*(R+2*r)*X(35)+2*(R+r)*(R-2*r)*X(36)

X(14804) lies on these lines: {1,3}, {79,6906}, {140,5427}, {411,5441}, {499,6901}, {1478,6888}, {1749,5694}, {3874,4996}, {4189,14450}, {4257,6126}, {5251,11681}, {5267,13407}, {5442,6940}, {6839,7741}, {6852,7951}, {6895,10483}

X(14804) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 5172, 14795), (3, 56, 3336), (36, 3746, 11012), (10966, 11021, 5049), (14796, 14797, 14794), (14801, 14802, 65)

X(14805) = CONJUGATE-COUPLE OF X(50) IN THE RANGE IN INVOLUTION {X(3), X(6)}, {X(15), X(16)}

Trilinears         (2*cos(2*A)+4)*cos(B-C)-cos(A)-2*cos(3*A) : :
Barycentrics    (SB+SC)*(3*(3*R^2-SW)*SA-S^2) : :
X(14805) = 3*(3*R^2-SW)*X(3)-SW*X(6)

X(14805) lies on these lines: {2,265}, {3,6}, {4,7712}, {5,10546}, {30,14389}, {49,7503}, {74,5012}, {140,12022}, {184,399}, {353,6235}, {381,1495}, {549,3580}, {1154,11004}, {1656,13367}, {1658,13434}, {2937,11424}, {3091,5944}, {3292,11935}, {3526,12038}, {3628,11449}, {3851,10282}, {5640,7575}, {5663,11003}, {5891,9703}, {5907,9704}, {5946,10298}, {7526,11456}, {7527,12112}, {7687,10254}, {8546,11579}, {9545,11591}, {9781,12107}, {9818,10540}, {10545,12106}, {11799,13394}

X(14805) = Brocard circle-inverse-of-X(3581)
X(14805) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 3431, 1511), (3, 6, 3581), (3, 567, 568), (3, 578, 6243), (3, 13352, 13340), (6, 3581, 568), (15, 16, 566), (39, 13351, 10898), (567, 3581, 6), (3311, 12239, 6421), (5092, 10564, 3), (5092, 11430, 10564)

X(14806) = CONJUGATE-COUPLE OF X(52) IN THE RANGE IN INVOLUTION {X(3), X(6)}, {X(15), X(16)}

Trilinears         (4*cos(A)*cos(B-C)+cos(2*A)+2)*sin(A) : :
Barycentrics    (2*R^2-3*SA-SW)*(SB+SC) : :
X(14806) = 2*SW*(R^2-2*SW)*X(6)-3*(S^2-SW^2)*X(32)

X(14806) lies on these lines: {2,1879}, {3,6}, {231,3523}, {233,7756}, {427,3055}, {631,2165}, {3054,7499}, {3815,7667}

X(14806) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 570, 571), (15, 16, 569), (39, 8553, 13345), (574, 10979, 6), (800, 6478, 9557), (1609, 5013, 5421)

X(14807) = ANTICOMPLEMENT OF X(1113)

Barycentrics    (3*R-OH)*SB*SC-R*S^2 : :
X(14807) = 3*X(2)-4*X(1313)

As a point on the Euler line, X(14807) has Shinagawa coefficeints (-R, 3*R-OH)

X(14807) lies on the anticomplementary circle and these lines: {2,3}, {10,2100}, {98,13581}, {110,14499}, {145,2102}, {146,2575}, {149,10781}, {193,2104}, {516,2101}, {1503,8116}, {2574,3448}, {2593,12384}

X(14807) = reflection of X(i) in X(j) for these (i,j): (2, 10719), (20, 1114), (145, 2102), (193, 2104), (1113, 1313), (2100, 10), (3146, 10736), (14808, 4)
X(14807) = an intersection of Euler line and anticomplementary circle
X(14807) = anticomplement of X(1113)
X(14807) = antipode of X(14808) in anticomplementary circle
X(14807) = orthocentroidal circle-inverse-of-X(2553)
X(14807) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 4, 2553), (423, 7540, 3091), (1009, 7513, 6970), (1113, 1313, 2), (1113, 6644, 297), (1113, 10719, 1313), (1325, 3861, 6660), (3855, 13632, 11314), (4241, 6888, 2070), (6990, 11325, 14002), (7520, 14033, 27), (11096, 14787, 436), (11299, 11308, 10995)

X(14808) = ANTICOMPLEMENT OF X(1114)

Barycentrics    (3*R+OH)*SB*SC-R*S^2 : :
X(14808) = 3*X(2)-4*X(1312)

As a point on the Euler line, X(14808) has Shinagawa coefficeints (-R, 3*R+OH)

X(14808) lies on the anticomplementary circle and these lines: {2,3}, {10,2101}, {98,13580}, {110,14500}, {145,2103}, {146,2574}, {193,2105}, {516,2100}, {1503,8115}, {2575,3448}, {2592,12384}

X(14808) = reflection of X(i) in X(j) for these (i,j): (2, 10720), (20, 1113), (145, 2103), (193, 2105), (1114, 1312), (2101, 10), (3146, 10737), (14807, 4)
X(14808) = an intersection of Euler line and anticomplementary circle
X(14808) = antipode of X(14807) in anticomplementary circle
X(14808) = orthocentroidal circle-inverse-of-X(2552)
X(14808) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 4, 2552), (3, 1346, 2), (20, 3153, 14807), (23, 7391, 14807), (401, 7575, 13745), (858, 1113, 2), (1114, 1312, 2), (1114, 10720, 1312), (3147, 7496, 1982), (3147, 11312, 14043), (3152, 11911, 10154), (5189, 10720, 2552), (8360, 11289, 8365)

X(14809) =  X(3)X(523)∩X(526)X(12041)

Barycentrics    a^2 (b-c) (b+c) (a^10-4 a^8 b^2+6 a^6 b^4-4 a^4 b^6+a^2 b^8-4 a^8 c^2+8 a^6 b^2 c^2-6 a^4 b^4 c^2+4 a^2 b^6 c^2-2 b^8 c^2+6 a^6 c^4-6 a^4 b^2 c^4-3 a^2 b^4 c^4+2 b^6 c^4-4 a^4 c^6+4 a^2 b^2 c^6+2 b^4 c^6+a^2 c^8-2 b^2 c^8) : :

X(14809) lies on the cubic K920 and these lines: {3,523}, {526,12041}, {1640,11063}, {11270,14380}

X(14810) =  X(3)X(6)∩X(4)X(7937)

Barycentrics    a^2 (2 a^4+a^2 b^2-3 b^4+a^2 c^2-4 b^2 c^2-3 c^4) : :
X(14810) = 5 X(3) - X(6) = 3 X(6) - 5 X(182) = 3 X(3) - X(182) = 4 X(6) - 5 X(575) = 4 X(182) - 3 X(575) = 4 X(3) - X(575) = 7 X(6) - 5 X(576) = 7 X(575) - 4 X(576) = 7 X(182) - 3 X(576) = 7 X(3) - X(576) = 3 X(3) + X(1350) = 3 X(575) + 4 X(1350) = 3 X(6) + 5 X(1350) = 3 X(576) + 7 X(1350) = 9 X(576) - 7 X(1351) = 9 X(6) - 5 X(1351) = 9 X(575) - 4 X(1351) = 9 X(3) - X(1351) = 3 X(182) - X(1351) = 3 X(1350) + X(1351) = 3 X(376) + X(1352) = X(1350) - 3 X(3098) = X(182) + 3 X(3098) = X(575) + 4 X(3098) = X(6) + 5 X(3098) = X(576) + 7 X(3098) = X(1351) + 9 X(3098) = X(69) + 7 X(3528) = X(3529) + 7 X(3619) = X(382) - 5 X(3763) = 11 X(6) - 15 X(5050) = 11 X(575) - 12 X(5050) = 11 X(182) - 9 X(5050) = 11 X(3) - 3 X(5050) = 11 X(3098) + 3 X(5050) = 11 X(1350) + 9 X(5050) = 7 X(6) - 15 X(5085) = 7 X(575) - 12 X(5085) = 7 X(5050) - 11 X(5085) = 7 X(182) - 9 X(5085) = 7 X(3) - 3 X(5085) = X(576) - 3 X(5085) = 7 X(3098) + 3 X(5085) = 7 X(1350) + 9 X(5085) = 6 X(5050) - 11 X(5092) = 2 X(1351) - 9 X(5092) = 2 X(576) - 7 X(5092) = 6 X(5085) - 7 X(5092) = 2 X(6) - 5 X(5092) = 2 X(182) - 3 X(5092) = 2 X(3098) + X(5092) = 2 X(1350) + 3 X(5092) = 19 X(6) - 15 X(5093) = 19 X(575) - 12 X(5093)

X(14810) lies on the cubic K920 and these lines: {2,14488}, {3,6}, {4,7937}, {20,3818}, {22,3819}, {23,5650}, {69,3528}, {74,12074}, {110,3917}, {141,550}, {159,3357}, {186,12294}, {373,7496}, {376,1352}, {382,3763}, {384,12122}, {542,8703}, {548,1503}, {549,5480}, {698,7780}, {733,11654}, {755,1296}, {1368,6723}, {1469,5010}, {1495,7492}, {1657,10516}, {1843,3520}, {2916,12162}, {2979,11003}, {3056,7280}, {3066,6688}, {3329,9751}, {3522,5921}, {3523,14561}, {3524,5476}, {3529,3619}, {3530,3589}, {3534,11178}, {3618,10299}, {3787,10329}, {5020,5646}, {5031,7842}, {5447,7525}, {5907,10323}, {5943,7485}, {6000,8717}, {6248,7470}, {6776,10304}, {7509,13598}, {7516,10110}, {7526,9822}, {7768,12252}, {7804,10007}, {8547,8681}, {9019,12039}, {10168,12100}, {11592,12107}, {11898,14093}, {14134,14135}

X(14810) = midpoint of X(i) and X(j) for these {i,j}: {3, 3098}, {20, 3818}, {74, 12584}, {141, 550}, {159, 3357}, {182, 1350}, {3534, 11178}
X(14810) = reflection of X(i) in X(j) for these {i,j}: {575, 5092}, {3589, 3530}, {5092, 3}, {5097, 182}, {10168, 12100}
X(14810) = Schoutte-circle-inverse of X(5008)
X(14810) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 1350, 182), (3, 5188, 13335), (3, 12305, 12975), (3, 12306, 12974), (3, 14538, 13349), (3, 14539, 13350), (15, 16, 5008), (182, 3098, 1350), (182, 5097, 575), (5052, 8589, 5116), (5092, 5097, 182), (5104, 5116, 5052), (5447, 7525, 10282), (7492, 7998, 1495), (8160, 8161, 5188), (13349, 13350, 13335), (14538, 14539, 5188)

X(14811) =  X(3)X(523)∩X(526)X(12041)

Barycentrics    a^2 (a^12 b^2-a^10 b^4-2 a^8 b^6+2 a^6 b^8+a^4 b^10-a^2 b^12+a^12 c^2-6 a^10 b^2 c^2+5 a^8 b^4 c^2-3 a^6 b^6 c^2+3 a^4 b^8 c^2-a^10 c^4+5 a^8 b^2 c^4+12 a^6 b^4 c^4-12 a^4 b^6 c^4+4 a^2 b^8 c^4-3 b^10 c^4-2 a^8 c^6-3 a^6 b^2 c^6-12 a^4 b^4 c^6+3 b^8 c^6+2 a^6 c^8+3 a^4 b^2 c^8+4 a^2 b^4 c^8+3 b^6 c^8+a^4 c^10-3 b^4 c^10-a^2 c^12) : :

X(14811) lies on the cubic K920 and these lines: {3,67}, {182,691}, {512,5092}

X(14812) =  X(1)X(513)∩X(100)X(109)

Barycentrics    a (b-c) (3 a^3-2 a^2 b-3 a b^2+2 b^3-2 a^2 c+5 a b c-b^2 c-3 a c^2-b c^2+2 c^3) : :

X(14812) lies on the cubic K921 and these lines: {1,513}, {58,3737}, {89,4724}, {100,109}, {145,522}, {390,6006}, {514,4644}, {521,2136}, {649,4266}, {900,7972}, {944,3667}, {953,2718}, {2827,10698}, {3309,7966}, {3751,9001}

X(14812) = crossdifference of every pair of points on line {44, 2170}
X(14812) = X(2718)-zayin conjugate of X(14513)

X(14813) =  1st GHIOCAS-LOZADA-EULER POINT

Trilinears    sqrt(3) cos A + 2 cos B cos C : :
X(14813) = sqrt(3)*X(3)+X(4) = X(4)-3*X(2044)

See Antreas Hatzipolakis and César Lozada, Hyacinthos 26655.

X(14813) lies on these lines: {2, 3}, {15, 3071}, {16, 3070}, {17, 615}, {18, 590}, {371, 398}, {372, 397}, {395, 8960}, {623, 642}, {624, 641}, {1151, 5339}, {1152, 5340}, {1587, 11486}, {1588, 11485}, {5318, 6396}, {5321, 6200}, {5334, 6221}, {5335, 6398}, {5343, 6449}, {5344, 6450}, {5365, 6455}, {5366, 6456}, {8976, 11489}, {8981, 11543}, {11488, 13951}, {11542, 13966}

X(14813) = midpoint of X(2041) and X(3529)
X(14813) = reflection of X(14814) in X(550)
X(14813) = complement of X(2042)
X(14813) = anticomplement of X(35738)
X(14813) = X(3519)-Ceva conjugate of X(14814)
X(14813) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2,1656,14814), (2, 2041, 632), (3, 4, 14814), (3, 382, 2043), (4, 2046, 5), (5,140,14814), (20,1657,14814), (376, 5073, 14814), (381,3523,14814), (382, 3522, 14814), (472, 1593, 4185), (631,3851,14814), (1884, 1904, 9714), (2567, 6861, 24), (3530, 3858, 14814), (3843, 10299, 14814), (5576, 7495, 14814), (7524, 13737, 7412)

X(14814) =  2nd GHIOCAS-LOZADA-EULER POINT

Trilinears    -sqrt(3) cos A + 2 cos B cos C : :
X(14814) = -sqrt(3) X(3) + X(4) = X(4) - 3 X(2043)

See Antreas Hatzipolakis and César Lozada, Hyacinthos 26655.

X(14814) lies on these lines: {2, 3}, {15, 3070}, {16, 3071}, {17, 590}, {18, 615}, {371, 397}, {372, 398}, {396, 8960}, {623, 641}, {624, 642}, {1151, 5340}, {1152, 5339}, {1587, 11485}, {1588, 11486}, {5318, 6200}, {5321, 6396}, {5334, 6398}, {5335, 6221}, {5343, 6450}, {5344, 6449}, {5365, 6456}, {5366, 6455}, {8976, 11488}, {8981, 11542}, {11489, 13951}, {11543, 13966}

X(14814) = midpoint of X(2042) and X(3529)
X(14814) = reflection of X(14813) in X(550)
X(14814) = complement of X(2041)
X(14814) = X(3519)-Ceva conjugate of X(14813)
X(14814) = radical-circle-of-Stammler-circles-inverse of X(7734)
X(14814) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2,1656,14813), (3, 4, 14813), (3, 382, 2044), (4, 2045, 5), (5,140,14813), (20,1657,14813), (376, 5073, 14813), (381, 3523, 14813), (382, 3522, 14813), (631,3851,14813), (2041, 2043, 3), (2041, 2045, 4), (3134, 13737, 6983), (3135, 3546, 6913), (3530, 3858, 14813), (3843, 10299, 14813), (5576, 7495, 14813)

leftri

Ghoicas-Lozada Images: X(14815)-X(14825)

rightri

This preamble was developed by Peter Moses and Clark Kimberling, October 11, 2017.

The following material is based on a construction by M. D. Ghoicas in 1934; see Antreas Hatzipolakis and César Lozada, Hyacinthos 26655. Let P be a point in the plane of a triangle ABC, not on one of the three sidelines, and let P* be the isogonal conjugate P*. Let

AB = AP∩BC, and define BC and CA cyclically.
AC = AP*∩BC, and define BA and CB cyclically.
A' = BBC /\ CCB, and define B' and C' cyclically.
A'' = BBA /\ CCA, and define B'' and C'' cyclically.

Then the following pairs of triangles are perspective: ABC and A'B'C', ABC and A''B''C'', and A'B'C' and A''B''C''. Let

Q = ABC and A'B'C'
Q* = perspector of ABC and A'B'C'
GL(P) = perspector of A'B'C' and A''B''C''

The point GL(P) is here named the Ghoicas-Lozada image of P. Let

U = angle BAP = angle CAP*,    V = angle CBP = angle ACP*,   

W = angle ACP = angle BAP*

Homogeneous trilinear coordinates for Q, Q*, and G(L,P) are as follows:

Q = (sin A)/sin(A - U) : (sin B)/sin(B - U) : (sin C)/sin(C - U) (cf. TCCT p.177, and much earlier: A. G. Burgess, "Concurrency of lines joining vertices of a triangle to opposite vewrtices of triangles on its side," Proceedings of the Edinburgh Mathematical Society 33 (1913-1914) 58-64; especialy p. 60.)

Q* = sin(A - U)/sin A : sin(B - U)/sin B : sin(C - U)/sin C

GL(P) = sin U sin(B - V) sin(C - W) + sin V sin W sin(A - U) : sin V sin(C - W) sin(A - U) + sin W sin U sin(B - U) : sin W sin(A - U) sin(B - V) + sin U sin V sin(C - V)

If X is a triangle center, then GL(X) is a triangle center.

If P= p : q : r (barycentrics), then GL(P) = a2p(c2q2 + b2r2) : :

If X = x : y : z (trilinears), then GL(X) = x(y2 + z2) : :

GL(P) = crosspoint of P and the isogonal conjugate of P
GL(P) = crosssum of P and the isogonal conjugate of P

The appearance of (i,j) in the following list means that G(X(i)) = X(j):

(1,1), (2,39), (3,185), (4,185), (6,39), (9,2082), (10,14815), (13,14816), (14,14817), (17,14818), (18,14819), (19,2083), (31,2085), (32,14820), (37,6155), (38,14821), (39,14822), (43,14823), (57,2082), (63,2083), (75,2085), (81,6155), (98,446), (99,14824), (101,14825), (100,6161), (511,446), (513,6161)

X(14815) =  GHIOCAS-LOZADA IMAGE OF X(10)

Barycentrics    a^2 (b+c) (a^2 b^2+2 a b^3+b^4+a^2 c^2+2 a c^3+c^4) : :

X(14815) lies on these lines: {1, 3122}, {5, 3120}, {31, 2915}, {58, 2126}, {72, 3778}, {238, 11102}, {256, 1010}, {859, 1193}, {986, 4425}, {1245, 1400}, {2292, 4205}, {3670, 3846}, {3725, 10974}, {4443, 9534}

X(14815) = crosspoint of X(10) and X(58)
X(14815) = crosssum of X(10) and X(58)

X(14816) =  GHIOCAS-LOZADA IMAGE OF X(13)

Barycentrics    (SB+SC)*(S+sqrt(3)*SA)*(sqrt(3)*(S^2+SB*SC)+(12*R^2-SA-SW)*S) : :

X(14816) lies on these lines: {3,6}, {30,8014}, {74,2981}, {5318,11555}, {5946,8016}, {11092,11303}

X(14816) = crosspoint of X(13) and X(15)
X(14816) = crosssum of X(13) and X(15)
X(14816) = {X(39),X(9730)}-harmonic conjugate of X(14817)

X(14817) =  GHIOCAS-LOZADA IMAGE OF X(14)

Barycentrics    (SB+SC)*(-S+sqrt(3)*SA)*(sqrt(3)*(S^2+SB*SC)-(12*R^2-SA-SW)*S) : :

X(14817) lies on these lines: {3,6}, {30,8015}, {74,6151}, {5321,11556}, {5946,8017}, {11078,11304}

X(14817) = crosspoint of X(14) and X(16)
X(14817) = crosssum of X(14) and X(16)
X(14817) = {X(39),X(9730)}-harmonic conjugate of X(14816)

X(14818) =  GHIOCAS-LOZADA IMAGE OF X(17)

Barycentrics    (SB+SC)*(3*S^2+sqrt(3)*(SA+SW)*S+(2*R^2+2*SA-SW)*SA) : :

X(14818) lies on these lines: {15,2927}, {49,61}, {54,2981}, {3107,3526}

X(14818) = crosspoint of X(17) and X(61)
X(14818) = crosssum of X(17) and X(61)

X(14819) =  GHIOCAS-LOZADA IMAGE OF X(18)

Barycentrics    (SB+SC)*(3*S^2-sqrt(3)*(SA+SW)*S+(2*R^2+2*SA-SW)*SA) : :

X(14819) lies on these lines: {16,2928}, {49,62}, {54,6151}, {3106,3526}

X(14819) = crosspoint of X(18) and X(62)
X(14819) = crosssum of X(18) and X(62)

X(14820) =  GHIOCAS-LOZADA IMAGE OF X(32)

Barycentrics    a^4 (b^2+c^2) (b^4-b^2 c^2+c^4) : :

X(14820) lies on these lines: {3, 8569}, {5, 3124}, {39, 3118}, {185, 446}, {384, 694}, {1084, 9490}, {3094, 7876}, {3981, 5025}, {8041, 8362}, {10339, 12212}

X(14820) = crosspoint of X(32) and X(76)
X(14820) = crosssum of X(32) and X(76)
X(14820) = {X(694),X(695)}-harmonic conjugate of X(384)
X(14820) = barycentric product X(i)*X(j) for these {i,j}: {39, 3981}, {3051, 5025}
X(14820) = barycentric quotient X(3981)/X(308)

X(14821) =  GHIOCAS-LOZADA IMAGE OF X(38)

Barycentrics    a (b^2+c^2) (2 a^4+2 a^2 b^2+b^4+2 a^2 c^2+c^4) : :

X(14821) lies on this line: {11709, 14933}

X(14821) = crosspoint of X(38) and X(82)
X(14821) = crosssum of X(38) and X(82)
X(14821) = barycentric product X(38)*X(7829)
X(14821) = barycentric quotient X(7829)/X(3112)

X(14822) =  GHIOCAS-LOZADA IMAGE OF X(39)

Barycentrics    a^2 (b^2+c^2) (a^4 b^2+a^4 c^2+4 a^2 b^2 c^2+b^4 c^2+b^2 c^4) : :

X(14822) lies on these lines: {2, 3499}, {3, 8570}, {6, 2896}, {39, 3118}, {597, 9490}, {1207, 7859}, {2086, 7829}, {3051, 8362}, {7935, 10014}

X(14822) = crosspoint of X(39) and X(83)
X(14822) = crosssum of X(39) and X(83)

X(14823) =  GHIOCAS-LOZADA IMAGE OF X(43)

Barycentrics    a (a b+a c-b c) (a^2 b^2-2 a^2 b c+a^2 c^2+b^2 c^2) : :

X(14823) lies on the cubic K258 and these lines: {1, 668}, {2, 3223}, {3, 238}, {5, 5518}, {43, 2176}, {386, 3993}

X(14823) = X(932)-Ceva conjugate of X(4083)
X(14823) = crosspoint of X(43) and X(87)
X(14823) = crosssum of X(43) and X(87)

X(14824) =  GHIOCAS-LOZADA IMAGE OF X(99)

Barycentrics    a^2 (b^2-c^2) (a^4 b^2+a^4 c^2-4 a^2 b^2 c^2+b^4 c^2+b^2 c^4) : :

X(14824) lies on these lines: {1, 9421}, {3, 669}, {39, 512}, {99, 2142}, {194, 523}, {690, 887}, {2134, 3733}, {2396, 11123}, {3566, 9491}, {8665, 10097}

X(14824) = crossdifference of every pair of points on line {385, 3291}
X(14824) = crosspoint of X(99) and X(512)
X(14824) = crosssum of X(99) and X(512)
X(14824) = 2nd-Brocard-circle-inverse of X(5108)
X(14824) = barycentric product X(647)*X(5186)
X(14824) = barycentric quotient X(5186)/X(6331)

X(14825) =  GHIOCAS-LOZADA IMAGE OF X(101)

Barycentrics    a^2 (-b+c) (a^2 b^2-2 a b^3+b^4+a^2 c^2-2 a c^3+c^4) : :

X(14825) lies on these lines: {1, 1024}, {3, 649}, {6, 8578}, {65, 650}, {85, 514}, {218, 657}, {392, 3250}, {442, 661}, {513, 1212}, {652, 3215}, {663, 2440}, {1642, 6161}, {1919, 2172}, {2176, 6586}, {3294, 4079}

X(14825) = crossdifference of every pair of points on line {1936, 3011}
X(14825) = crosspoint of X(101) and X(514)
X(14825) = crosssum of X(101) and X(514)
X(14825) = barycentric product X(4025)X(5185)
X(14825) = barycentric quotient X(5185)/X(1897)

X(14826) =  X(2)X(98)∩X(25)X(69)

Barycentrics    3*a^6 - 3*a^4*b^2 + a^2*b^4 - b^6 - 3*a^4*c^2 + 6*a^2*b^2*c^2 + b^4*c^2 + a^2*c^4 + b^2*c^4 - c^6 : :

X(14826) lies on the cubic K173 and these lines: {2, 98}, {3, 11206}, {4, 394}, {5, 3167}, {6, 7392}, {20, 3917}, {22, 10519}, {25, 69}, {49, 14786}, {51, 193}, {66, 1660}, {76, 6620}, {141, 154}, {155, 7401}, {237, 3785}, {323, 7394}, {343, 6353}, {427, 6090}, {486, 8970}, {487, 3155}, {488, 3156}, {511, 6995}, {631, 3796}, {638, 5200}, {1092, 3088}, {1147, 7404}, {1181, 6803}, {1353, 10128}, {1495, 3620}, {1498, 10996}, {1503, 7386}, {1583, 12257}, {1584, 12256}, {1614, 7383}, {1851, 1943}, {1944, 7102}, {1975, 4176}, {1992, 9777}, {1993, 6997}, {1995, 6515}, {2979, 7500}, {3087, 3289}, {3091, 3292}, {3148, 3926}, {3332, 6817}, {3522, 13445}, {3539, 10783}, {3540, 10784}, {3547, 10539}, {3564, 5020}, {3618, 11402}, {3619, 7499}, {3787, 7737}, {3818, 7378}, {4198, 10441}, {5449, 12420}, {5462, 9936}, {5544, 13361}, {5562, 7487}, {6146, 6804}, {6524, 9308}, {6642, 11411}, {6643, 12134}, {6676, 8780}, {6759, 7400}, {6815, 11441}, {6816, 14516}, {7396, 11550}, {8889, 11064}, {8968, 10515}, {9703, 14787}, {9825, 12164}, {9833, 11793}, {10132, 11292}, {10133, 11291}, {11245, 11284}, {12162, 12250}
X(14826) = reflection of X(11433) in X(5020)

X(14826) = pivot of the cubic K173
X(14826) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 5921, 1899), (5, 3167, 11427), (20, 11444, 11821), (141, 154, 7494), (193, 7398, 51), (1352, 9306, 2), (1899, 5651, 2)

X(14827) =  X(6)X(1174)∩X(31)X(32)

Barycentrics    a^4*(a - b - c)^2 : :
Barycentrics    Cot(A/2)^2 Sin(A)^4 : :

A'B'C' and A"B"C" be the Mandart-incircle and anti-Mandart-incircle triangles, resp. Let A* be the barycentric product A'*A", and define B* and C* cyclically. The lines AA**, BB**, CC** concur in X(14827). (Randy Hutson, November 2, 2017)

X(14827) lies on these lines: {6, 1174}, {31, 32}, {39, 7124}, {48, 5065}, {55, 2195}, {101, 9306}, {220, 1260}, {294, 1621}, {574, 7117}, {577, 2174}, {607, 1500}, {902, 1200}, {949, 6184}, {1190, 3052}, {1253, 6602}, {1397, 9454}, {1407, 3207}, {1951, 2242}, {2082, 2241}, {2175, 9447}, {7109, 9449}, {8735, 9664}

X(14827) = X(i)-Ceva conjugate of X(j) for these (i,j): {41, 2175}, {1262, 692}
X(14827) = crosspoint of X(i) and X(j) for these (i,j): {41, 1253}, {220, 7071}, {692, 1262}
X(14827) = crossdifference of every pair of points on line {693, 6362}
X(14827) = barycentric square of X(55)
X(14827) = X(i)-isoconjugate of X(j) for these (i,j): {2, 1088}, {7, 85}, {57, 6063}, {69, 1847}, {75, 279}, {76, 269}, {77, 331}, {86, 1446}, {92, 7056}, {264, 7177}, {273, 348}, {274, 3668}, {278, 7182}, {304, 1119}, {305, 1435}, {309, 14256}, {310, 1427}, {312, 479}, {349, 1014}, {514, 4569}, {523, 4635}, {552, 6358}, {555, 4146}, {561, 1407}, {658, 693}, {670, 7216}, {738, 3596}, {850, 4637}, {873, 6354}, {934, 3261}, {1042, 6385}, {1106, 1502}, {1111, 1275}, {1432, 7205}, {1434, 1441}, {1577, 4616}, {1969, 7053}, {3212, 7209}, {3669, 4572}, {3676, 4554}, {4025, 13149}, {4077, 4573}, {4391, 4626}, {4566, 7199}, {4602, 7250}, {4625, 7178}, {7196, 7249}, {7233, 10030}
X(14827) = crosssum of X(i) and X(j) for these (i,j): {2, 6604}, {6, 1602}, {85, 1088}, {279, 7056}, {693, 1146}
X(14827) = barycentric product X(i)*X(j) for these {i,j}: {1, 1253}, {3, 7071}, {6, 220}, {8, 2175}, {9, 41}, {11, 6066}, {19, 1802}, {25, 1260}, {31, 200}, {32, 346}, {33, 212}, {42, 2328}, {48, 7079}, {55, 55}, {56, 480}, {57, 6602}, {59, 3022}, {60, 7064}, {71, 2332}, {78, 2212}, {100, 8641}, {101, 657}, {109, 4105}, {110, 4524}, {163, 4171}, {181, 6061}, {184, 7046}, {198, 7367}, {210, 2194}, {213, 2287}, {219, 607}, {228, 4183}, {284, 1334}, {312, 9447}, {341, 560}, {604, 728}, {644, 3063}, {663, 3939}, {667, 4578}, {669, 7256}, {692, 3900}, {798, 7259}, {872, 1098}, {1043, 1918}, {1110, 2310}, {1170, 8551}, {1174, 8012}, {1259, 6059}, {1265, 1974}, {1333, 4515}, {1397, 5423}, {1415, 4130}, {1436, 7368}, {1500, 7054}, {1857, 6056}, {1919, 6558}, {1924, 7258}, {1973, 3692}, {2149, 3119}, {2192, 7074}, {2195, 2340}, {2200, 2322}, {2204, 3694}, {2206, 4082}, {2293, 10482}, {2299, 2318}, {2324, 7118}, {2327, 2333}, {3271, 6065}, {3596, 9448}, {3709, 5546}, {4319, 7084}, {6064, 7063}, {6559, 9454}, {7058, 7109}, {7101, 9247}
X(14827) = barycentric quotient X(i)/X(j) for these {i,j}: {31, 1088}, {32, 279}, {41, 85}, {55, 6063}, {163, 4635}, {184, 7056}, {200, 561}, {212, 7182}, {213, 1446}, {220, 76}, {341, 1928}, {346, 1502}, {480, 3596}, {560, 269}, {607, 331}, {657, 3261}, {692, 4569}, {1253, 75}, {1260, 305}, {1334, 349}, {1397, 479}, {1501, 1407}, {1576, 4616}, {1802, 304}, {1917, 1106}, {1918, 3668}, {1924, 7216}, {1973, 1847}, {1974, 1119}, {2175, 7}, {2205, 1427}, {2212, 273}, {2287, 6385}, {2328, 310}, {2330, 7205}, {3939, 4572}, {4524, 850}, {4578, 6386}, {6056, 7055}, {6066, 4998}, {6602, 312}, {7063, 1365}, {7071, 264}, {7079, 1969}, {7109, 6354}, {7256, 4609}, {7259, 4602}, {8012, 1233}, {8551, 1229}, {8641, 693}, {9247, 7177}, {9426, 7250}, {9447, 57}, {9448, 56}, {14575, 7053}

X(14828) = X(1)X(85)∩X(2)X(6)

Barycentrics    a^4 - a^3*b - a^2*b^2 + a*b^3 - a^3*c - 3*a^2*b*c - a*b^2*c + b^3*c - a^2*c^2 - a*b*c^2 - 2*b^2*c^2 + a*c^3 + b*c^3 : :

X(14828) lies on these lines: {1, 85}, {2, 6}, {3, 1434}, {7, 55}, {37, 10025}, {75, 3870}, {77, 1088}, {142, 3684}, {150, 495}, {171, 3664}, {200, 10436}, {226, 4872}, {286, 14004}, {348, 5703}, {350, 1233}, {354, 1447}, {673, 2280}, {894, 3693}, {1509, 4592}, {1565, 5719}, {2646, 7176}, {3550, 4888}, {3598, 11038}, {3663, 3750}, {3879, 4847}, {3957, 4360}, {4038, 11019}, {4209, 4258}, {4386, 4675}, {4911, 13407}, {7580, 10446}

X(14828) = crossdifference of every pair of points on line {512, 10581}
X(14828) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 3945, 14548), (3664, 13405, 9436), (3945, 5736, 86)

X(14829) =  X(2)X(6)∩X(8)X(56)

Barycentrics    a^3 - a*b^2 + a*b*c - b^2*c - a*c^2 - b*c^2 : :

X(14829) lies on these lines: {1, 3769}, {2, 6}, {3, 1043}, {5, 1330}, {8, 56}, {11, 4388}, {27, 264}, {40, 4673}, {43, 2274}, {55, 10453}, {57, 75}, {58, 13740}, {63, 190}, {76, 6996}, {78, 7572}, {95, 306}, {100, 3996}, {149, 4450}, {165, 3886}, {171, 3741}, {210, 5205}, {226, 320}, {238, 3840}, {239, 3752}, {314, 1764}, {315, 7377}, {317, 469}, {319, 2985}, {321, 3218}, {322, 3306}, {344, 5273}, {345, 5744}, {354, 3757}, {474, 9534}, {518, 7081}, {519, 4256}, {553, 7321}, {638, 2048}, {658, 7182}, {662, 7058}, {673, 2319}, {757, 14534}, {908, 4001}, {911, 4586}, {982, 4362}, {999, 5774}, {1010, 10479}, {1089, 6763}, {1155, 3706}, {1220, 1468}, {1230, 1232}, {1465, 1943}, {1707, 4676}, {1724, 13741}, {1792, 6986}, {1834, 4201}, {1999, 3666}, {2000, 4123}, {2050, 10446}, {2329, 3912}, {2886, 4645}, {2999, 3759}, {3187, 4850}, {3219, 4358}, {3227, 8056}, {3262, 4359}, {3286, 4203}, {3336, 4647}, {3416, 3705}, {3452, 4416}, {3662, 3772}, {3681, 3699}, {3685, 4640}, {3686, 6692}, {3729, 3928}, {3831, 5247}, {3883, 11019}, {3891, 4392}, {3916, 7283}, {3923, 4650}, {3944, 4655}, {3961, 4434}, {4011, 7262}, {4042, 4413}, {4195, 4252}, {4234, 4257}, {4384, 5437}, {4415, 6646}, {4649, 6685}, {4684, 13405}, {4720, 13587}, {4966, 6690}, {4981, 5297}, {5015, 10916}, {6172, 8055}, {6327, 11680}, {6734, 7270}, {6999, 7750}, {7413, 10477}

X(14829) = isotomic conjugate of X(2051)
X(14829) = anticomplement of X(37662)
X(14829) = X(4564)-Ceva conjugate of X(190)
X(14829) = X(572)-cross conjugate of X(11109)
X(14829) = cevapoint of X(2) and X(1764)
X(14829) = crosspoint of X(799) and X(4998)
X(14829) = crosssum of X(798) and X(3271)
X(14829) = X(4560)-zayin conjugate of X(798)
X(14829) = crosspoint, wrt excentral or anticomplementary triangle, of X(2) and X(1764)
X(14829) = barycentric product X(i)*X(j) for these {i,j}: {69, 11109}, {75, 2975}, {76, 572}
X(14829) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 2051}, {572, 6}, {2975, 1}, {11109, 4}, {11998, 2170}
X(14829) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 69, 4417), (2, 940, 86), (2, 1150, 333), (2, 1654, 5743), (2, 2895, 5741), (2, 5361, 5278), (2, 5372, 1150), (2, 5739, 5233), (2, 14552, 14555), (3, 10449, 1043), (8, 12513, 1222), (57, 11679, 75), (63, 312, 190), (171, 3741, 5263), (491, 492, 5224), (1150, 5278, 5361), (1999, 3666, 4360), (5278, 5361, 333)

X(14830) =  X(3)X(67)∩X(30)X(98)

Barycentrics    5*a^8 - 6*a^6*b^2 + 5*a^4*b^4 - 3*a^2*b^6 - b^8 - 6*a^6*c^2 - a^4*b^2*c^2 + 2*a^2*b^4*c^2 + 5*a^4*c^4 + 2*a^2*b^2*c^4 + 2*b^4*c^4 - 3*a^2*c^6 - c^8 : :
X(14830) = 3 X(98) - X(671), 3 X(3) - 2 X(2482), X(147) - 3 X(3524), 2 X(114) - 3 X(5054), 3 X(381) - 4 X(5461), 3 X(5055) - 4 X(6036), 2 X(5461) - 3 X(6055), 4 X(671) - 3 X(6321), 4 X(98) - X(6321), 3 X(376) - X(8591), 3 X(20) + X(8596), 4 X(2482) - 3 X(8724), 2 X(3845) - 3 X(9166), X(6033) + 2 X(9862), 3 X(8182) - X(9890), X(5984) + 3 X(10304), 3 X(9166) - X(10722), 7 X(6321) - 4 X(10723), 7 X(671) - 3 X(10723), 7 X(98) - X(10723), X(3) + 2 X(10991), X(2482) + 3 X(10991), X(8724) + 4 X(10991), X(8591) + 3 X(11177), 2 X(8787) - 3 X(11179), X(382) - 4 X(11623), 2 X(10723) - 7 X(11632), 2 X(671) - 3 X(11632), X(6033) - 4 X(12042), X(9862) + 2 X(12042), X(8596) - 3 X(12243), 4 X(5066) - 5 X(14061), X(3543) - 3 X(14651), 6 X(10304) - X(14692), 2 X(5984) + X(14692)

X(14830) lies on the cubics K728 ande K905 and these lines: {2, 5191}, {3, 67}, {4, 8587}, {20, 8596}, {30, 98}, {32, 6034}, {35, 12350}, {36, 12351}, {99, 8703}, {114, 5054}, {115, 1384}, {147, 3524}, {148, 11001}, {187, 11645}, {376, 2782}, {381, 2794}, {382, 9880}, {524, 9142}, {530, 6295}, {531, 6582}, {543, 3534}, {549, 6054}, {550, 12117}, {591, 9894}, {597, 1576}, {1499, 14443}, {1657, 12355}, {1991, 9892}, {1992, 3095}, {2777, 11656}, {2793, 14666}, {2796, 8669}, {3058, 10069}, {3543, 14651}, {3579, 9881}, {3582, 12185}, {3584, 12184}, {3656, 11710}, {3793, 10754}, {3845, 9166}, {3972, 10033}, {4302, 12354}, {5024, 5477}, {5055, 6036}, {5066, 14061}, {5152, 7811}, {5182, 8359}, {5434, 10053}, {5465, 7728}, {5476, 11842}, {5969, 9821}, {5984, 10304}, {6222, 12602}, {6284, 10070}, {6287, 13335}, {6308, 8725}, {6399, 12601}, {7354, 10054}, {7739, 12829}, {7771, 9774}, {7793, 9878}, {7840, 8295}, {8182, 9830}, {8593, 11155}, {8787, 11171}, {9143, 9155}, {9855, 10810}, {11006, 12041}, {12258, 12699}, {13663, 13749}, {13665, 13908}, {13748, 13783}, {13785, 13968}

X(14830) = midpoint of X(i) and X(j) for these {i,j}: {2, 9862}, {20, 12243}, {148, 11001}, {376, 11177}, {1657, 12355}, {3534, 12188}
X(14830) = circumcircle-inverse of X(32305)
X(14830) = reflection of X(i) in X(j) for these {i,j}: {2, 12042}, {99, 8703}, {381, 6055}, {382, 9880}, {3656, 11710}, {3830, 115}, {6033, 2}, {6054, 549}, {6321, 11632}, {7728, 5465}, {8724, 3}, {9880, 11623}, {9881, 3579}, {10722, 3845}, {11006, 12041}, {11632, 98}, {12117, 550}, {12699, 12258}
X(14830) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (9166, 10722, 3845), (9862, 12042, 6033)

X(14831) =  X(2)X(389)∩X(30)X(52)

Barycentrics    a^2*(3*a^6*b^2 - 9*a^4*b^4 + 9*a^2*b^6 - 3*b^8 + 3*a^6*c^2 - 2*a^4*b^2*c^2 - 5*a^2*b^4*c^2 + 4*b^6*c^2 - 9*a^4*c^4 - 5*a^2*b^2*c^4 - 2*b^4*c^4 + 9*a^2*c^6 + 4*b^2*c^6 - 3*c^8) : :
X(14831) = 2 X(52) + X(185) = 3 X(51) - 2 X(381) = 9 X(373) - 8 X(547) = X(381) - 3 X(568) = 3 X(3060) - X(3543) = 3 X(3545) - 5 X(3567) = 4 X(549) - 3 X(3917) = 2 X(1216) - 3 X(5054) = 3 X(5055) - 4 X(5462) = 4 X(389) - X(5562) = 2 X(389) + X(5889) = X(5562) + 2 X(5889) = X(376) - 3 X(5890) = 4 X(547) - 3 X(5891) = 3 X(373) - 2 X(5891) = 3 X(5650) - 4 X(5892) = 3 X(3545) - 2 X(5907) = 5 X(3567) - 2 X(5907) = 5 X(5071) - 6 X(5943) = 3 X(373) - 4 X(5946) = 2 X(547) - 3 X(5946) = X(185) - 4 X(6102) = X(52) + 2 X(6102) = 3 X(3524) - 4 X(9729) = 2 X(549) - 3 X(9730) = 3 X(3839) - 4 X(10110) = X(6240) + 2 X(10112) = 5 X(52) - 2 X(10263) = 5 X(6102) + X(10263) = 5 X(185) + 4 X(10263) = 3 X(10304) - 5 X(10574) = 5 X(185) - 2 X(10575) = 10 X(6102) - X(10575) = 5 X(52) + X(10575) = 2 X(10263) + X(10575) = 4 X(5446) - X(11381) = 3 X(3524) - X(11412) = 4 X(9729) - X(11412) = 7 X(5562) - 10 X(11444) = 7 X(2) - 5 X(11444) = 14 X(389) - 5 X(11444) = 7 X(5889) + 5 X(11444) = 5 X(5071) - 3 X(11459) = 7 X(5562) - 16 X(11695) = 7 X(2) - 8 X(11695) = 5 X(11444) - 8 X(11695) = 7 X(389) - 4 X(11695) = 7 X(5889) + 8 X(11695) = 5 X(5562) - 8 X(11793) = 10 X(11695) - 7 X(11793) = 5 X(2) - 4 X(11793)

X(14831) lies on these lines: {2, 389}, {3, 13366}, {20, 13382}, {30, 52}, {51, 381}, {143, 3845}, {184, 14070}, {195, 12038}, {373, 547}, {376, 511}, {378, 576}, {541, 13417}, {542, 1843}, {549, 1154}, {1092, 9786}, {1181, 9909}, {1216, 5054}, {1351, 10605}, {1568, 13567}, {1993, 11438}, {1994, 11430}, {3060, 3543}, {3292, 6644}, {3520, 13482}, {3524, 9729}, {3534, 6243}, {3545, 3567}, {3574, 12359}, {3830, 5446}, {3839, 10110}, {5055, 5462}, {5066, 5876}, {5071, 5943}, {5076, 12002}, {5102, 10606}, {5498, 11803}, {5650, 5892}, {5655, 11557}, {6101, 12100}, {6240, 10112}, {6241, 13598}, {7722, 11800}, {8703, 10625}, {9140, 10628}, {10298, 11422}, {10304, 10574}, {11225, 12022}, {11424, 12163}, {11539, 12006}, {12161, 13367}, {13340, 14093}

X(14831) = midpoint of X(i) and X(j) for these {i,j}: {2, 5889}, {3534, 6243}
X(14831) = reflection of X(i) in X(j) for these {i,j}: {2, 389}, {51, 568}, {3830, 5446}, {3845, 143}, {3917, 9730}, {5562, 2}, {5655, 11557}, {5876, 5066}, {5891, 5946}, {6101, 12100}, {8703, 13630}, {10625, 8703}, {11381, 3830}, {11459, 5943}, {12022, 11225}, {12162, 3845}
X(14831) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (52, 6102, 185), (52, 10575, 10263), (389, 5889, 5562), (5891, 5946, 373), (9786, 12160, 1092)

X(14832) =  REFLECTION OF X(10418) IN X(115)

Barycentrics    2*a^10 - 4*a^8*b^2 - 3*a^6*b^4 + 8*a^4*b^6 + 2*a^2*b^8 - 3*b^10 - 4*a^8*c^2 + 18*a^6*b^2*c^2 - 12*a^4*b^4*c^2 - 25*a^2*b^6*c^2 + 12*b^8*c^2 - 3*a^6*c^4 - 12*a^4*b^2*c^4 + 48*a^2*b^4*c^4 - 9*b^6*c^4 + 8*a^4*c^6 - 25*a^2*b^2*c^6 - 9*b^4*c^6 + 2*a^2*c^8 + 12*b^2*c^8 - 3*c^10 : :

X(14832) lies on the cubic K870 and these lines: {2, 99}, {5477, 9144}

X(14832) = reflection of X(10418) in X(115)

X(14833) =  X(4)X(542)∩X(110)X(9830)

Barycentrics    (a^2 + b^2 - 2*c^2)*(a^2 - 2*b^2 + c^2)*(5*a^8 - 6*a^6*b^2 + 2*a^4*b^4 - 3*a^2*b^6 + 2*b^8 - 6*a^6*c^2 + 5*a^4*b^2*c^2 + 2*a^2*b^4*c^2 + 2*a^4*c^4 + 2*a^2*b^2*c^4 - 4*b^4*c^4 - 3*a^2*c^6 + 2*c^8) : :

X(14833) lies on the cubic K870 and these lines: {4, 542}, {110, 9830}, {111, 1648}, {690, 11161}, {691, 11645}, {5181, 8591}, {5465, 8593}, {6593, 10488}, {6776, 11656}

X(14833) = reflection of X(i) in X(j) for these {i,j}: {895, 671}, {6776, 11656}, {8591, 5181}, {8593, 5465}, {9140, 11646}, {10488, 6593}, {13169, 11161}
X(14833) = reflection of X(11161) in the Fermat line
X(14833) = X(671)-daleth conjugate of X(1992)

X(14834) =  X(2)X(98)∩X(8288)X(14644)

Barycentrics    a^16 - a^14*b^2 - 3*a^12*b^4 + 10*a^10*b^6 - 13*a^8*b^8 + a^6*b^10 + 13*a^4*b^12 - 10*a^2*b^14 + 2*b^16 - a^14*c^2 + 4*a^12*b^2*c^2 - 6*a^10*b^4*c^2 - 9*a^8*b^6*c^2 + 40*a^6*b^8*c^2 - 57*a^4*b^10*c^2 + 39*a^2*b^12*c^2 - 10*b^14*c^2 - 3*a^12*c^4 - 6*a^10*b^2*c^4 + 37*a^8*b^4*c^4 - 40*a^6*b^6*c^4 + 54*a^4*b^8*c^4 - 50*a^2*b^10*c^4 + 12*b^12*c^4 + 10*a^10*c^6 - 9*a^8*b^2*c^6 - 40*a^6*b^4*c^6 - 20*a^4*b^6*c^6 + 21*a^2*b^8*c^6 + 10*b^10*c^6 - 13*a^8*c^8 + 40*a^6*b^2*c^8 + 54*a^4*b^4*c^8 + 21*a^2*b^6*c^8 - 28*b^8*c^8 + a^6*c^10 - 57*a^4*b^2*c^10 - 50*a^2*b^4*c^10 + 10*b^6*c^10 + 13*a^4*c^12 + 39*a^2*b^2*c^12 + 12*b^4*c^12 - 10*a^2*c^14 - 10*b^2*c^14 + 2*c^16 : :

X(14834) lies on these lines: {2, 98}, {8288, 14644}

X(14835) =  X(1)X(9271)∩X(678)X(1023)

Barycentrics    a*(b - c)^2*(-2*a + b + c)^4 : :

X(14835) lies on these lines: {1, 9271}, {678, 1023}, {2087, 3251}

X(14835) = X(679)-Ceva conjugate of X(1635)
X(14835) = crosspoint of X(679) and X(1635)
X(14835) = crosssum of X(i) and X(j) for these (i,j): {100, 9326}, {678, 3257}
X(14835) = barycentric product X(i)*X(j) for these {i,j}: {2087, 8028}, {3251, 6544}
X(14835) = barycentric quotient X(14637)/X(1022)

X(14836) =  X(5)X(1989)∩X(6)X(30)

Barycentrics    4*a^8 - a^6*b^2 - 9*a^4*b^4 + 5*a^2*b^6 + b^8 - a^6*c^2 - 2*a^4*b^2*c^2 - 5*a^2*b^4*c^2 - 4*b^6*c^2 - 9*a^4*c^4 - 5*a^2*b^2*c^4 + 6*b^4*c^4 + 5*a^2*c^6 - 4*b^2*c^6 + c^8 : :

X(14836) lies on the cubic K173 and these lines: {5, 1989}, {6, 30}, {39, 3163}, {50, 8703}, {427, 1990}, {523, 14582}, {549, 566}, {1194, 3003}, {3018, 3815}, {5158, 5309}, {5254, 6128}

X(14836) = X(14389)-Ceva conjugate of X(5)
X(14836) = crossdifference of every pair of points on line {3581, 8675}
X(14836) = crosssum of X(6) and X(4550)

X(14837) = COMPLEMENT OF X(6332)

Barycentrics    (b - c)*(-a^3 - a^2*b + a*b^2 + b^3 - a^2*c + 2*a*b*c - b^2*c + a*c^2 - b*c^2 + c^3) : :
X(14837) = 3 X(1638) - X(3669) = 3 X(4453) + X(4462) = 2 X(4521) + X(4707) = 2 X(7658) + X(10015)

X(14837) lies on these lines: {2, 2399}, {10, 4163}, {57, 4091}, {240, 522}, {241, 514}, {525, 3239}, {676, 3900}, {1751, 4049}, {2401, 8056}, {2457, 4778}, {2490, 7657}, {3667, 7655}, {3798, 6002}, {3910, 4885}, {4025, 4391}, {4040, 8713}, {4147, 4458}, {4453, 4462}, {4521, 4707}, {6129, 8058}

X(14837) = midpoint of X(i) and X(j) for these {i,j}: {650, 7178}, {656, 7649}, {905, 10015}, {2490, 7657}, {4025, 4391}, {4147, 4458}
X(14837) = reflection of X(i) in X(j) for these {i,j}: {905, 7658}, {4163, 10}
X(14837) = isogonal conjugate of X(36049)
X(14837) = complement X(6332)
X(14837) = X(i)-Ceva conjugate of X(j) for these (i,j): {7, 3318}, {653, 196}, {4025, 522}, {4391, 514}, {4617, 3663}
X(14837) = X(i)-cross conjugate of X(j) for these (i,j): {3318, 7}, {5514, 7952}, {14298, 8058}
X(14837) = cevapoint of X(i) and X(j) for these (i,j): {656, 6587}, {6129, 14298}
X(14837) = crosspoint of X(i) and X(j) for these (i,j): {2, 653}, {75, 658}
X(14837) = crossdifference of every pair of points on line {48, 55}
X(14837) = crosssum of X(i) and X(j) for these (i,j): {6, 652}, {31, 657}, {650, 1108}
X(14837) = X(i)-aleph conjugate of X(j) for these (i,j): {509, 2636}, {653, 522}
X(14837) = X(i)-beth conjugate of X(j) for these (i,j): {8, 4163}, {333, 6332}
X(14837) = X(77)-gimel conjugate of X(4091)
X(14837) = X(i)-zayin conjugate of X(j) for these (i,j): {34, 1783}, {108, 652}, {513, 579}, {676, 8558}, {1119, 1020}, {1459, 1743}, {1461, 657}, {3676, 1708}, {4341, 651}, {7649, 1723}, {7661, 1741}
X(14837) = trilinear pole of line X(3318)X(6087)
X(14837) = X(i)-complementary conjugate of X(j) for these (i,j): {6, 123}, {19, 124}, {34, 116}, {65, 127}, {108, 141}, {112, 960}, {604, 2968}, {607, 5514}, {608, 11}, {651, 1368}, {653, 2887}, {1395, 1086}, {1398, 4904}, {1409, 122}, {1415, 3}, {1783, 1329}, {1880, 125}, {1973, 1146}, {2203, 4858}, {2207, 6506}, {2212, 13609}, {3209, 7358}, {7012, 3835}, {7115, 513}, {8750, 3452}
X(14837) = X(i)-isoconjugate of X(j) for these (i,j): {6, 13138}, {9, 8059}, {84, 101}, {100, 1436}, {108, 268}, {109, 282}, {110, 1903}, {189, 692}, {190, 2208}, {280, 1415}, {285, 4559}, {644, 1413}, {651, 2192}, {653, 2188}, {662, 2357}, {664, 7118}, {934, 7367}, {1331, 7129}, {1332, 7151}, {1422, 3939}, {1433, 1783}, {1490, 8064}, {1813, 7008}, {2182, 6081}, {4578, 6612}, {6516, 7154}
X(14837) = barycentric product X(i)*X(j) for these {i,j}: {7, 8058}, {40, 693}, {75, 6129}, {85, 14298}, {196, 6332}, {198, 3261}, {223, 4391}, {322, 513}, {329, 514}, {331, 10397}, {342, 521}, {347, 522}, {523, 8822}, {658, 5514}, {850, 2360}, {1577, 1817}, {3194, 14208}, {3239, 14256}, {3676, 7080}, {4025, 7952}
X(14837) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 13138}, {40, 100}, {56, 8059}, {102, 6081}, {196, 653}, {198, 101}, {208, 108}, {221, 109}, {223, 651}, {227, 4551}, {322, 668}, {329, 190}, {347, 664}, {512, 2357}, {513, 84}, {514, 189}, {521, 271}, {522, 280}, {649, 1436}, {650, 282}, {652, 268}, {657, 7367}, {661, 1903}, {663, 2192}, {667, 2208}, {693, 309}, {1459, 1433}, {1817, 662}, {1946, 2188}, {2187, 692}, {2199, 1415}, {2324, 644}, {2331, 1783}, {2360, 110}, {3063, 7118}, {3064, 7003}, {3194, 162}, {3195, 8750}, {3318, 8058}, {3669, 1422}, {3676, 1440}, {3737, 285}, {6087, 515}, {6129, 1}, {6591, 7129}, {6611, 1461}, {7011, 1813}, {7013, 6516}, {7074, 3939}, {7078, 1331}, {7080, 3699}, {7152, 8064}, {7178, 8808}, {7952, 1897}, {8058, 8}, {8822, 99}, {10397, 219}, {14256, 658}, {14298, 9}

X(14838) =  COMPLEMENT OF X(1557)

Barycentrics    a*(b - c)*(a^2 - b^2 - b*c - c^2) : :
X(14838) = 3 X(905) - X(3669) = 3 X(650) + X(3669) = 2 X(3669) - 3 X(3960) = 2 X(650) + X(3960) = 3 X(1635) - X(4063) = 3 X(2) + X(4560) = 7 X(3624) - X(4804) = 2 X(1125) + X(4913) = X(4367) - 3 X(14419) = X(4705) + 3 X(14419)

X(14838) lies on these lines: {1, 4041}, {2, 1577}, {6, 7254}, {10, 3907}, {116, 7117}, {241, 514}, {249, 662}, {386, 810}, {512, 9508}, {513, 4401}, {522, 8062}, {523, 8043}, {647, 8045}, {649, 14349}, {656, 3737}, {657, 4091}, {659, 2530}, {661, 1019}, {663, 1734}, {667, 830}, {758, 2614}, {759, 842}, {784, 4874}, {798, 4481}, {824, 6586}, {899, 9980}, {1125, 4151}, {1575, 9321}, {1635, 4063}, {2254, 4040}, {2516, 8712}, {2522, 3798}, {2523, 4932}, {2526, 3803}, {2611, 9213}, {2786, 3709}, {2832, 3777}, {3287, 8774}, {3309, 4794}, {3624, 4804}, {3716, 8714}, {4129, 6002}, {4160, 4367}, {4164, 9256}, {4378, 4490}, {4467, 7265}, {4724, 4905}, {4730, 4879}, {4784, 4983}, {4823, 4885}, {6362, 6675}, {6710, 13006}

X(14838) = midpoint of X(i) and X(j) for these {i,j}: {1, 4041}, {649, 14349}, {650, 905}, {656, 3737}, {659, 2530}, {661, 1019}, {663, 1734}, {667, 1491}, {798, 4481}, {1577, 4560}, {2254, 4040}, {2526, 3803}, {4367, 4705}, {4378, 4490}, {4467, 7265}, {4724, 4905}, {4730, 4879}, {4784, 4983}
X(14838) = reflection of X(i) in X(j) for these {i,j}: {3960, 905}, {4401, 6050}, {4823, 4885}
X(14838) = isotomic conjugate of X(15455)
X(14838) = complement X(1577)
X(14838) = X(i)-complementary conjugate of X(j) for these (i,j): {3, 127}, {6, 125}, {31, 8287}, {32, 115}, {50, 3258}, {58, 116}, {99, 626}, {101, 3454}, {110, 141}, {112, 5}, {163, 10}, {187, 5099}, {248, 3150}, {249, 512}, {251, 7668}, {284, 124}, {571, 136}, {577, 122}, {604, 8286}, {662, 2887}, {691, 625}, {692, 1211}, {805, 5031}, {827, 3934}, {933, 14767}, {1101, 4369}, {1110, 4129}, {1333, 11}, {1379, 2040}, {1380, 2039}, {1384, 5512}, {1408, 4904}, {1415, 442}, {1501, 1084}, {1576, 2}, {1609, 135}, {1625, 1209}, {1691, 2679}, {1918, 6627}, {1970, 130}, {1974, 6388}, {2193, 123}, {2204, 6506}, {2206, 1086}, {2420, 113}, {2715, 511}, {2965, 137}, {3053, 5139}, {3285, 3259}, {4556, 3741}, {4558, 1368}, {4565, 2886}, {4570, 3835}, {4630, 3589}, {5467, 126}, {5546, 1329}, {5994, 624}, {5995, 623}, {7953, 7849}, {10547, 339}, {11060, 10413}, {13345, 5522}, {14533, 2972}, {14560, 3580}, {14574, 39}, {14586, 140}, {14591, 1511}, {14642, 13611}, {14838,21253}
X(14838) = X(i)-Ceva conjugate of X(j) for these (i,j): {2, 8287}, {7, 3024}, {1414, 1}, {3219, 7202}, {4077, 6003}, {4608, 513}, {6578, 3743}, {7372, 523}
X(14838) = X(i)-cross conjugate of X(j) for these (i,j): {3024, 7}, {7202, 3219}
X(14838) = X(i)-isoconjugate of X(j) for these (i,j): {6, 6742}, {37, 13486}, {79, 101}, {100, 2160}, {109, 7110}, {110, 8818}, {163, 6757}, {190, 6186}, {476, 2245}, {651, 7073}, {653, 8606}, {1783, 7100}, {1983, 2166}, {3615, 4559}, {3936, 14560}
X(14838) = cevapoint of X(2605) and X(9404)
X(14838) = crosspoint of X(i) and X(j) for these (i,j): {2, 662}, {651, 1255}
X(14838) = trilinear pole of line X(526)X(2611)
X(14838) = trilinear square root of X(7266)
X(14838) = crossdifference of every pair of points on line {55, 199}
X(14838) = crosssum of X(i) and X(j) for these (i,j): {6, 661}, {650, 1100}, {657, 14547}
X(14838) = polar conjugate of isogonal conjugate of X(23226)
X(14838) = X(i)-aleph conjugate of X(j) for these (i,j): {81, 5400}, {99, 6002}, {266, 5539}, {509, 2640}, {662, 6003}, {4573, 4369}, {14089, 6002}
X(14838) = X(333)-beth conjugate of X(1577)
X(14838) = X(i)-zayin conjugate of X(j) for these (i,j): {7, 651}, {56, 101}, {110, 661}, {1019, 4253}, {1804, 1461}, {3737, 6}, {4017, 1743}, {7203, 4383}, {9811, 896}, {10571, 1783}
X(14838) = X(i)-isoconjugate of X(j) for these (i,j): {6, 6742}, {37, 13486}, {79, 101}, {100, 2160}, {109, 7110}, {110, 8818}, {163, 6757}, {190, 6186}, {476, 2245}, {651, 7073}, {653, 8606}, {1783, 7100}, {1983, 2166}, {3615, 4559}, {3936, 14560}
X(14838) = barycentric product X(i)*X(j) for these {i,j}: {1, 4467}, {35, 693}, {75, 2605}, {81, 7265}, {85, 9404}, {99, 2611}, {190, 7202}, {319, 513}, {514, 3219}, {521, 7282}, {522, 1442}, {526, 14616}, {662, 8287}, {759, 3268}, {1019, 3969}, {1414, 6741}, {2003, 4391}, {2174, 3261}, {3647, 4608}, {3676, 4420}, {3678, 7192}, {4025, 6198}, {6742, 7266}
X(14838) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 6742}, {35, 100}, {50, 1983}, {58, 13486}, {186, 4242}, {319, 668}, {323, 4585}, {513, 79}, {523, 6757}, {526, 758}, {649, 2160}, {650, 7110}, {661, 8818}, {663, 7073}, {667, 6186}, {759, 476}, {1399, 109}, {1442, 664}, {1459, 7100}, {1946, 8606}, {2003, 651}, {2088, 2610}, {2174, 101}, {2594, 4551}, {2605, 1}, {2611, 523}, {2624, 2245}, {3219, 190}, {3647, 4427}, {3678, 3952}, {3737, 3615}, {3969, 4033}, {4420, 3699}, {4467, 75}, {6198, 1897}, {6741, 4086}, {7186, 3888}, {7202, 514}, {7265, 321}, {7266, 4467}, {8287, 1577}, {9404, 9}, {14270, 3724}
X(14838) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 4560, 1577), (4705, 14419, 4367)

X(14839) =  X(1)X(39)∩X(8)X(76)

Barycentrics    a*(a^2*b^2 - a*b^3 - a*b^2*c + a^2*c^2 - a*b*c^2 + 2*b^2*c^2 - a*c^3) : :

X(14839) lies on these lines: {1, 39}, {2, 3799}, {3, 8671}, {6, 10800}, {8, 76}, {10, 3934}, {30, 511}, {38, 4475}, {40, 5188}, {65, 7198}, {75, 3688}, {99, 3110}, {100, 5091}, {101, 8301}, {105, 644}, {120, 4904}, {145, 194}, {187, 11364}, {190, 3271}, {262, 5603}, {355, 6248}, {384, 12195}, {664, 1362}, {944, 11257}, {984, 1573}, {1001, 4752}, {1002, 3227}, {1017, 8297}, {1086, 4553}, {1125, 6683}, {1146, 3041}, {1280, 10699}, {1385, 13334}, {1386, 3997}, {1482, 3095}, {1572, 3751}, {1916, 3903}, {2023, 11725}, {2098, 12836}, {2099, 12837}, {2170, 4712}, {3030, 3699}, {3056, 3729}, {3094, 3242}, {3099, 5184}, {3616, 7786}, {3632, 9902}, {3679, 3789}, {3680, 9442}, {3779, 3875}, {3809, 4850}, {3888, 4014}, {3913, 12338}, {3971, 5943}, {4111, 5564}, {4384, 4517}, {4488, 9309}, {4492, 4792}, {4677, 14711}, {5007, 12194}, {5008, 10789}, {5790, 7697}, {5901, 11272}, {6272, 12628}, {6273, 12627}, {7709, 7967}, {7804, 10791}, {8992, 13911}, {9821, 12702}, {9884, 11152}, {9917, 12410}, {9983, 12495}, {10063, 12647}, {10079, 10573}, {10246, 11171}, {10912, 12923}, {10950, 13077}, {12135, 12143}, {12245, 12251}, {12454, 12474}, {12455, 12475}, {12626, 12794}, {12635, 12933}, {12636, 12992}, {12637, 12993}, {12645, 13108}, {12648, 13109}, {12649, 13110}, {13973, 13983}

X(14839) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 291, 1015), (1, 1018, 8299), (1, 12782, 39), (10, 12263, 3934), (105, 644, 1083), (145, 194, 7976), (2007, 2008, 13006), (3888, 4440, 4014), (4919, 9451, 1)
X(14839) = isogonal conjugate of X(14665)
X(14839) = crossdifference of every pair of points on line {6, 659}

X(14840) =  (name pending)

Barycentrics    (2*a^2 - 3*a*b + 2*b^2 - 2*c^2)*(2*a^2 + 3*a*b + 2*b^2 - 2*c^2)*(2*a^2 - 2*b^2 - 3*a*c + 2*c^2)*(2*a^2 - 2*b^2 + 3*a*c + 2*c^2) : :
Barycentrics    Sin(A)^2/(8 Cos(2 A) - 1) : :

X(14840) lies on the circumconic {A, B, C, X(2), X(6)}.

X(14841) =  CEVAPOINT OF X(1656) AND X(1657)

Barycentrics    (2*a^2 - 3*a*b + 2*b^2 - 2*c^2)*(2*a^2 + 3*a*b + 2*b^2 - 2*c^2)*(a^2 - b^2 - c^2)*(2*a^2 - 2*b^2 - 3*a*c + 2*c^2)*(2*a^2 - 2*b^2 + 3*a*c + 2*c^2) : :
Barycentrics    Cos(A) Sin(A)/(8 Cos(2 A) - 1) : :

X(14841) lies on the Jerabek hyhperbola and these lines: {1173, 3851}, {1656, 13472}, {1657, 13452}

X(14841) = cevapoint of X(1656) and X(1657)

X(14842) =  (name pending)

Barycentrics    (3*a^2 - 4*a*b + 3*b^2 - 3*c^2)*(3*a^2 + 4*a*b + 3*b^2 - 3*c^2)*(3*a^2 - 3*b^2 - 4*a*c + 3*c^2)*(3*a^2 - 3*b^2 + 4*a*c + 3*c^2) : :
Barycentrics    Sin(A)^2/(1 + 9 Cos(2 A)) : :

X(14842) lies on the circumconic {A, B, C, X(2), X(6)}

X(14843) =  CEVAPOINT OF X(3090) AND X(3529)

Barycentrics    (3*a^2 - 4*a*b + 3*b^2 - 3*c^2)*(3*a^2 + 4*a*b + 3*b^2 - 3*c^2)*(a^2 - b^2 - c^2)*(3*a^2 - 3*b^2 - 4*a*c + 3*c^2)*(3*a^2 - 3*b^2 + 4*a*c + 3*c^2) : :
Barycentrics    Cos(A) Sin(A)/(1 + 9 Cos(2 A)) : :

X(14843) lies on the Jerabek hyperbola and these lines: {64, 11541}, {3525, 14528}

X(14843) = cevapoint of X(3090) and X(3529)

X(14844) =  X(11)X(7073)∩X(21)X(36)

Barycentrics    (a^2 + a*b + b^2 - c^2)*(a^2 - a*b - b^2 - a*c - b*c - c^2)*(a^2 - b^2 + a*c + c^2) : :

X(14844) lies on these lines: {11, 7073}, {21, 36}, {502, 1255}

X(14844) = X(8818)-Ceva conjugate of X(79)
X(14844) = X(i)-isoconjugate of X(j) for these (i,j): {35, 13610}, {2174, 6625}, {2248, 3219}
X(14844) = barycentric product X(i)*X(j) for these {i,j}: {79, 1654}, {6626, 8818}
X(14844) = barycentric quotient X(i)/X(j) for these {i,j}: {79, 6625}, {846, 3219}, {1654, 319}, {2160, 13610}, {6186, 2248}

X(14845) =  X(3)X(6688)∩X(5)X(51)

Barycentrics    a^2*(a^2*b^2 - b^4 + a^2*c^2 + 2*b^2*c^2 - c^4)*(a^4 - 2*a^2*b^2 + b^4 - 2*a^2*c^2 - 8*b^2*c^2 + c^4) : :
X(14845) = 2 X(5) + X(51) = 4 X(51) - X(52) = 8 X(5) + X(52) = 7 X(52) - 16 X(143) = 7 X(51) - 4 X(143) = 7 X(5) + 2 X(143) = X(2979) - 7 X(3090) = 5 X(1656) - 2 X(3819) = X(185) + 8 X(3850) = 2 X(389) + 7 X(3851) = 4 X(547) - X(3917) = 2 X(1216) - 11 X(5056) = 5 X(3567) + 13 X(5068) = X(3060) + 5 X(5071) = X(2979) + 2 X(5446) = 7 X(3090) + 2 X(5446) = 13 X(5067) - 4 X(5447) = 5 X(3091) + 4 X(5462) = 10 X(5) - X(5562) = 5 X(51) + X(5562) = 5 X(52) + 4 X(5562) = 4 X(5462) - X(5890) = 5 X(3091) + X(5890) = 2 X(5562) - 5 X(5891) = 4 X(5) - X(5891) = 2 X(51) + X(5891) = X(52) + 2 X(5891) = 8 X(143) + 7 X(5891) = X(4) + 2 X(5892) = 11 X(5072) - 2 X(5907) = X(381) + 2 X(5943) = 2 X(5066) + X(5946) = 17 X(3854) + X(6241) = X(3) - 4 X(6688) = 5 X(3843) + 4 X(9729) = 4 X(5943) - X(9730) = 2 X(381) + X(9730) = 2 X(1216) + 7 X(9781) = 11 X(5056) + 7 X(9781) = X(9967) + 2 X(9971) = 5 X(143) - 14 X(10095) = 5 X(51) - 8 X(10095) = 5 X(5) + 4 X(10095) = X(5562) + 8 X(10095) = 5 X(5891) + 16 X(10095) = X(3819) + 2 X(10110) = 5 X(1656) + 4 X(10110) = 5 X(5071) - 2 X(10170) = X(3060) + 2 X(10170) = 8 X(546) + X(10575) = 10 X(1656) - X(10625) = 4 X(3819) - X(10625) = 8 X(10110) + X(10625)

X(14845) is the centroid of the nine points for which the nine-point circle is named. (Randy Hutson, November 2, 2017)

X(14845) lies on these lines: {3, 6688}, {4, 5892}, {5, 51}, {30, 373}, {113, 12099}, {154, 569}, {185, 3850}, {186, 10545}, {381, 1853}, {382, 11695}, {389, 3851}, {511, 5055}, {546, 10575}, {547, 3917}, {575, 10540}, {1216, 5056}, {1598, 13336}, {1656, 3819}, {2386, 13240}, {2393, 14561}, {2979, 3090}, {3060, 5071}, {3066, 9818}, {3091, 5462}, {3146, 11465}, {3153, 7693}, {3526, 13598}, {3545, 5640}, {3567, 5068}, {3830, 13570}, {3832, 11455}, {3843, 9729}, {3845, 13363}, {3854, 6241}, {3856, 13491}, {3857, 13630}, {3858, 11381}, {5020, 13352}, {5066, 5946}, {5067, 5447}, {5072, 5907}, {5079, 11793}, {5092, 5899}, {5650, 13391}, {6101, 12812}, {6102, 12811}, {7506, 11202}, {9967, 9971}, {10109, 13451}, {10539, 11402}

X(14845) = midpoint of X(3534) and X(5640)
X(14845) = QA-P26 (2nd QA-Quasi Centroid) of quadrangle ABCX(4)
{X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4, 11451, 5892), (5, 51, 5891), (5, 10095, 5562), (5, 13364, 51), (51, 5891, 52), (381, 5943, 9730), (1656, 10110, 10625), (3060, 5071, 10170), (3091, 5462, 12162), (3858, 12006, 11381), (5056, 9781, 1216)

X(14846) =  X(2)X(249)∩X(111)X(265)

Barycentrics    (a^2 - a*b + b^2 - c^2)*(a^2 + a*b + b^2 - c^2)*(a^2 - b^2 - a*c + c^2)*(a^2 - b^2 + a*c + c^2)*(a^10 - 2*a^8*b^2 + 4*a^6*b^4 - 8*a^4*b^6 + 7*a^2*b^8 - 2*b^10 - 2*a^8*c^2 - 2*a^6*b^2*c^2 + 6*a^4*b^4*c^2 - 5*a^2*b^6*c^2 + b^8*c^2 + 4*a^6*c^4 + 6*a^4*b^2*c^4 - 3*a^2*b^4*c^4 + b^6*c^4 - 8*a^4*c^6 - 5*a^2*b^2*c^6 + b^4*c^6 + 7*a^2*c^8 + b^2*c^8 - 2*c^10) : :

X(14846) lies on the Hutson-Parry circle and these lines: {2, 249}, {111, 265}, {476, 1648}, {1989, 9140}, {2433, 5627}, {5466, 14582}, {6792, 14356}

X(14847) =  X(4)-LINE CONJUGATE OF X(74)

Barycentrics    (2*a^4 - a^2*b^2 - b^4 - a^2*c^2 + 2*b^2*c^2 - c^4)*(2*a^14 - 3*a^12*b^2 - 5*a^10*b^4 + 9*a^8*b^6 + 4*a^6*b^8 - 13*a^4*b^10 + 7*a^2*b^12 - b^14 - 3*a^12*c^2 + 16*a^10*b^2*c^2 - 10*a^8*b^4*c^2 - 32*a^6*b^6*c^2 + 41*a^4*b^8*c^2 - 8*a^2*b^10*c^2 - 4*b^12*c^2 - 5*a^10*c^4 - 10*a^8*b^2*c^4 + 56*a^6*b^4*c^4 - 28*a^4*b^6*c^4 - 31*a^2*b^8*c^4 + 18*b^10*c^4 + 9*a^8*c^6 - 32*a^6*b^2*c^6 - 28*a^4*b^4*c^6 + 64*a^2*b^6*c^6 - 13*b^8*c^6 + 4*a^6*c^8 + 41*a^4*b^2*c^8 - 31*a^2*b^4*c^8 - 13*b^6*c^8 - 13*a^4*c^10 - 8*a^2*b^2*c^10 + 18*b^4*c^10 + 7*a^2*c^12 - 4*b^2*c^12 - c^14) : :
X(14847) = 2 X(107) + X(125), X(5667) + 2 X(7687), 4 X(133) - X(13202)

X(14847) lies on the cubic K173 and these lines: {4, 74}, {1636, 1637}, {3258, 11657}

X(14847) = crossdifference of every pair of points on line {74, 1636}
X(14847) = Dao-Moses-Telv-circle-inverse of X(3163)
X(14847) = X(4)-daleth conjugate of X(13202)
X(14847) = X(4)-Hirst inverse of X(5667)
X(14847) = X(i)-line conjugate of X(j) for these (i,j): {4, 74}, {1637, 1636}

X(14848)  =  X(2)X(1351)∩X(6)X(13)

Barycentrics    5*a^6 - 14*a^4*b^2 + 7*a^2*b^4 + 2*b^6 - 14*a^4*c^2 - 18*a^2*b^2*c^2 - 2*b^4*c^2 + 7*a^2*c^4 - 2*b^2*c^4 + 2*c^6 : :
X(14848) = 2 X(6) + X(381), X(69) - 4 X(547), X(382) + 8 X(575), X(3) - 4 X(597), 2 X(576) + X(599), 2 X(2) + X(1351), 2 X(599) - 5 X(1656), 4 X(576) + 5 X(1656), 2 X(5) + X(1992), 4 X(182) - X(3534), 2 X(549) - 5 X(3618), 7 X(381) - 4 X(3818), 7 X(6) + 2 X(3818), X(1353) + 2 X(5066), X(193) + 5 X(5071), X(3818) - 7 X(5476), X(381) - 4 X(5476), X(6) + 2 X(5476), X(3830) - 4 X(5480), 2 X(3845) + X(6776), 5 X(3843) + 4 X(8550), X(1352) + 2 X(8584), X(1350) - 4 X(10168), 4 X(10796) - X(11159), 7 X(3090) - X(11160), 2 X(5097) + X(11178), 2 X(5480) + X(11179), X(3830) + 2 X(11179), 4 X(5066) - X(11180), 2 X(1353) + X(11180), 7 X(3526) + 2 X(11477), 2 X(1992) - 5 X(11482), 4 X(5) + 5 X(11482), 4 X(11178) - X(11898), 8 X(5097) + X(11898), 2 X(376) - 5 X(12017), X(11188) - 4 X(13364), X(2104) + 2 X(13626), X(2105) + 2 X(13627), 8 X(5092) - 5 X(14093), X(5093) + 2 X(14561)

X(14848) lies on these lines: {2, 1351}, {3, 597}, {5, 1992}, {6, 13}, {30, 5050}, {69, 547}, {182, 3534}, {193, 5071}, {376, 12017}, {382, 575}, {403, 11405}, {511, 5054}, {524, 5055}, {549, 3618}, {576, 599}, {1350, 10168}, {1352, 8584}, {1353, 5066}, {1503, 14269}, {2104, 13626}, {2105, 13627}, {3066, 5642}, {3090, 11160}, {3526, 11477}, {3545, 3564}, {3830, 5480}, {3843, 8550}, {3845, 6776}, {5092, 14093}, {5097, 11178}, {5182, 10796}, {5969, 11165}, {8369, 10983}, {8976, 9975}, {9041, 10247}, {9974, 13951}, {10519, 11539}, {11184, 14645}, {11188, 13364}

X(14848) = midpoint of X(i) and X(j) for these {i,j}: {3545, 5032}, {5055, 5093}
X(14848) = reflection of X(i) in X(j) for these {i,j}: {5055, 14561}, {10519, 11539}
X(14848) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6, 5476, 381), (1353, 5066, 11180), (5480, 11179, 3830)

X(14849) =  X(98)X(265)∩X(115)X(7728)

Barycentrics    a^14 - a^12*b^2 - 2*a^10*b^4 + 4*a^8*b^6 - 6*a^6*b^8 + 8*a^4*b^10 - 5*a^2*b^12 + b^14 - a^12*c^2 + 2*a^10*b^2*c^2 + 7*a^6*b^6*c^2 - 16*a^4*b^8*c^2 + 14*a^2*b^10*c^2 - 6*b^12*c^2 - 2*a^10*c^4 - 9*a^6*b^4*c^4 + 9*a^4*b^6*c^4 - 17*a^2*b^8*c^4 + 12*b^10*c^4 + 4*a^8*c^6 + 7*a^6*b^2*c^6 + 9*a^4*b^4*c^6 + 16*a^2*b^6*c^6 - 7*b^8*c^6 - 6*a^6*c^8 - 16*a^4*b^2*c^8 - 17*a^2*b^4*c^8 - 7*b^6*c^8 + 8*a^4*c^10 + 14*a^2*b^2*c^10 + 12*b^4*c^10 - 5*a^2*c^12 - 6*b^2*c^12 + c^14 : :
X(14849) = 2 X(98) + X(265) = 4 X(115) - X(7728) = X(9862) + 2 X(10113) = X(148) + 2 X(12041) = 4 X(12042) - X(12121) = 2 X(125) + X(12188) = X(9860) + 2 X(12261) = 4 X(11710) - X(12898) = 4 X(6699) - X(13188)

X(14849) lies on these lines: {98, 265}, {115, 7728}, {125, 12188}, {148, 12041}, {542, 5050}, {690, 11632}, {5663, 14651}, {6699, 13188}, {9860, 12261}, {9862, 10113}, {11710, 12898}, {12042, 12121}

X(14850) =  X(3)X(67)∩X(99)X(265)

Barycentrics    a^14 - 5*a^12*b^2 + 6*a^10*b^4 + 4*a^8*b^6 - 14*a^6*b^8 + 12*a^4*b^10 - 5*a^2*b^12 + b^14 - 5*a^12*c^2 + 18*a^10*b^2*c^2 - 24*a^8*b^4*c^2 + 19*a^6*b^6*c^2 - 12*a^4*b^8*c^2 + 6*a^2*b^10*c^2 - 2*b^12*c^2 + 6*a^10*c^4 - 24*a^8*b^2*c^4 + 15*a^6*b^4*c^4 - 3*a^4*b^6*c^4 + 3*a^2*b^8*c^4 + 4*a^8*c^6 + 19*a^6*b^2*c^6 - 3*a^4*b^4*c^6 - 8*a^2*b^6*c^6 + b^8*c^6 - 14*a^6*c^8 - 12*a^4*b^2*c^8 + 3*a^2*b^4*c^8 + b^6*c^8 + 12*a^4*c^10 + 6*a^2*b^2*c^10 - 5*a^2*c^12 - 2*b^2*c^12 + c^14 : :
X(14850) = 2 X(99) + X(265) = 4 X(114) - X(7728) = X(5655) + 2 X(11006) = X(147) + 2 X(12041) = 2 X(11005) + X(12121) = 4 X(6699) - X(12188) = 4 X(11711) - X(12898) = 2 X(10113) + X(13172) = 2 X(12261) + X(13174) = 2 X(125) + X(13188)

X(14850) lies on these lines: {3, 67}, {99, 265}, {114, 7728}, {125, 13188}, {147, 12041}, {690, 14643}, {5655, 11006}, {6699, 12188}, {10113, 13172}, {11005, 12121}, {11711, 12898}, {12261, 13174}

X(14851) =  X(3)X(3447)∩X(30)X(14643)

Barycentrics    a^16 - a^14*b^2 - 10*a^12*b^4 + 29*a^10*b^6 - 30*a^8*b^8 + 9*a^6*b^10 + 6*a^4*b^12 - 5*a^2*b^14 + b^16 - a^14*c^2 + 18*a^12*b^2*c^2 - 25*a^10*b^4*c^2 - 17*a^8*b^6*c^2 + 42*a^6*b^8*c^2 - 23*a^4*b^10*c^2 + 12*a^2*b^12*c^2 - 6*b^14*c^2 - 10*a^12*c^4 - 25*a^10*b^2*c^4 + 87*a^8*b^4*c^4 - 50*a^6*b^6*c^4 - 12*a^4*b^8*c^4 - 6*a^2*b^10*c^4 + 16*b^12*c^4 + 29*a^10*c^6 - 17*a^8*b^2*c^6 - 50*a^6*b^4*c^6 + 58*a^4*b^6*c^6 - a^2*b^8*c^6 - 26*b^10*c^6 - 30*a^8*c^8 + 42*a^6*b^2*c^8 - 12*a^4*b^4*c^8 - a^2*b^6*c^8 + 30*b^8*c^8 + 9*a^6*c^10 - 23*a^4*b^2*c^10 - 6*a^2*b^4*c^10 - 26*b^6*c^10 + 6*a^4*c^12 + 12*a^2*b^2*c^12 + 16*b^4*c^12 - 5*a^2*c^14 - 6*b^2*c^14 + c^16 : :
X(14851) = X(265) + 2 X(477) = 4 X(3258) - X(7728) = 2 X(12041) + X(14731)

X(14851) lies on the cubic K173 and these lines: {3, 3447}, {30, 14643}, {265, 477}, {3258, 7728}, {12041, 14731}

X(14852) = X(3)X(125)∩X(5)X(6)

Barycentrics    (a^2 - b^2 - c^2)*(a^8 - a^6*b^2 + a^4*b^4 - 3*a^2*b^6 + 2*b^8 - a^6*c^2 + 2*a^4*b^2*c^2 + 3*a^2*b^4*c^2 - 8*b^6*c^2 + a^4*c^4 + 3*a^2*b^2*c^4 + 12*b^4*c^4 - 3*a^2*c^6 - 8*b^2*c^6 + 2*c^8) : :
X(14852) = 2 X(5) + X(68) = 4 X(5) - X(155) = 2 X(68) + X(155) = 2 X(1147) - 5 X(1656) = 2 X(265) + X(2931) = X(3167) - 3 X(5055) = 7 X(3851) - 4 X(5448) = X(3) - 4 X(5449) = 7 X(3090) - X(6193) = X(382) + 2 X(7689) = 7 X(3090) - 4 X(9820) = X(6193) - 4 X(9820) = 5 X(8227) + X(9896) = 2 X(5449) + X(9927) = X(3) + 2 X(9927) = 4 X(5901) - X(9933) = 5 X(155) - 2 X(9936) = 10 X(5) - X(9936) = 5 X(5654) - X(9936) = 5 X(68) + X(9936) = X(9928) - 4 X(9956) = 5 X(3091) + X(11411) = 7 X(3526) - 4 X(12038) = 4 X(140) - X(12118) = 2 X(4) + X(12163) = 7 X(3851) - X(12164) = 4 X(5448) - X(12164) = X(5562) + 2 X(12235) = X(355) + 2 X(12259) = 4 X(9927) - X(12293) = 2 X(3) + X(12293) = 8 X(5449) + X(12293) = 4 X(125) - X(12302) = X(12163) - 4 X(12359) = X(4) + 2 X(12359) = 2 X(1147) + X(12429) = 5 X(1656) + X(12429) = 2 X(12893) + X(12902) = X(9833) - 4 X(13383) = X(12084) - 4 X(13561)

X(14852) lies on these lines: {2, 12022}, {3, 125}, {4, 3580}, {5, 6}, {11, 10055}, {12, 10071}, {30, 1853}, {51, 381}, {52, 7507}, {113, 12165}, {140, 12118}, {143, 7564}, {154, 10201}, {355, 12259}, {382, 3581}, {394, 2072}, {403, 11442}, {498, 12428}, {539, 3167}, {567, 1147}, {912, 4654}, {1069, 7741}, {1181, 10024}, {1209, 7395}, {1350, 14791}, {1986, 12111}, {1993, 7577}, {2782, 14640}, {3090, 6193}, {3091, 11411}, {3157, 7951}, {3311, 13909}, {3312, 13970}, {3448, 11456}, {3526, 12038}, {3542, 12134}, {3549, 6146}, {3851, 5448}, {5094, 13352}, {5446, 12294}, {5562, 12235}, {5576, 10982}, {5876, 12236}, {5889, 7547}, {5901, 9933}, {6238, 10896}, {6288, 7506}, {6643, 10519}, {6800, 7552}, {6804, 12318}, {7352, 10895}, {7503, 9932}, {7505, 14516}, {7529, 9908}, {7565, 11002}, {7566, 9781}, {7569, 13434}, {8057, 14566}, {8227, 9896}, {8681, 11178}, {8909, 10576}, {9715, 11750}, {9833, 13383}, {9928, 9956}, {10104, 12193}, {10733, 11454}, {11425, 12370}, {11459, 14644}, {12084, 13561}

X(14852) = midpoint of X(68) and X(5654)
X(14852) = reflection of X(i) in X(j) for these {i,j}: {154, 10201}, {155, 5654}, {5654, 5}
X(14852) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 9927, 12293), (4, 12359, 12163), (5, 68, 155), (1656, 12429, 1147), (3090, 6193, 9820), (3851, 12164, 5448), (5449, 9927, 3)

X(14853) =  X(2)X(51)∩X(4)X(6)

Trilinears    sin A tan ω + cos B cos C : :
Barycentrics    (tan A)(tan B + tan C + cot ω) : :
Barycentrics    a^6 - 5*a^4*b^2 + 3*a^2*b^4 + b^6 - 5*a^4*c^2 - 6*a^2*b^2*c^2 - b^4*c^2 + 3*a^2*c^4 - b^2*c^4 + c^6 : :
X(14853) = X(4) + 2 X(6) = 4 X(5) - X(69) = X(20) - 4 X(182) = X(193) - 4 X(576) = X(376) - 4 X(597) = 2 X(113) + X(895) = 5 X(631) - 2 X(1350) = 2 X(5) + X(1351) = X(69) + 2 X(1351) = 2 X(576) + X(1352) = X(193) + 2 X(1352) = 2 X(546) + X(1353) = X(944) - 4 X(1386) = 2 X(381) + X(1992) = 2 X(1312) + X(2104) = 2 X(1313) + X(2105) = 4 X(141) - 7 X(3090) = 2 X(1352) - 5 X(3091) = X(193) + 5 X(3091) = 4 X(576) + 5 X(3091) = 8 X(575) + X(3146) = 4 X(3098) - 7 X(3523) = 5 X(631) - 8 X(3589) = X(1350) - 4 X(3589) = 2 X(3) - 5 X(3618) = 10 X(1656) - 7 X(3619) = 2 X(946) + X(3751) = 4 X(3818) - 7 X(3832) = 4 X(3629) + 11 X(3855) = X(1181) + 2 X(3867) = 5 X(3620) - 11 X(5056) = 10 X(3763) - 13 X(5067) = 2 X(599) - 5 X(5071) = 5 X(3522) - 8 X(5092) = X(3818) + 2 X(5097) = 7 X(3832) + 8 X(5097) = 3 X(3545) + 2 X(5102) = X(2) - 4 X(5476) = X(4) - 4 X(5480)

X(14853) lies on these lines: {2, 51}, {3, 3618}, {4, 6}, {5, 69}, {11, 10759}, {20, 182}, {25, 11427}, {30, 5050}, {52, 7404}, {54, 206}, {66, 1173}, {67, 14491}, {83, 13355}, {98, 5039}, {113, 895}, {114, 10754}, {115, 10753}, {116, 10758}, {117, 10764}, {118, 10756}, {119, 10755}, {124, 10757}, {125, 10752}, {132, 10766}, {133, 10762}, {141, 3090}, {146, 11579}, {148, 12177}, {154, 7714}, {159, 10594}, {184, 6995}, {193, 576}, {235, 12167}, {265, 11061}, {376, 597}, {381, 1992}, {388, 613}, {389, 3088}, {391, 7407}, {394, 7392}, {427, 9777}, {428, 11206}, {436, 6525}, {497, 611}, {518, 5603}, {524, 3545}, {542, 3839}, {546, 1353}, {550, 12017}, {569, 1176}, {575, 3146}, {578, 1974}, {599, 5071}, {631, 1350}, {944, 1386}, {946, 3751}, {966, 7380}, {1007, 6393}, {1131, 14231}, {1132, 14245}, {1205, 11807}, {1312, 2104}, {1313, 2105}, {1370, 5422}, {1428, 4293}, {1469, 3086}, {1513, 7736}, {1568, 9813}, {1570, 5475}, {1595, 11432}, {1596, 10602}, {1656, 3619}, {1691, 10788}, {1692, 7737}, {1843, 3089}, {1890, 2261}, {1899, 7378}, {1907, 12324}, {1993, 6997}, {1994, 7394}, {2330, 4294}, {2456, 10796}, {2548, 5028}, {2549, 5034}, {2777, 5622}, {3056, 3085}, {3066, 11064}, {3068, 6813}, {3069, 6811}, {3095, 3926}, {3098, 3523}, {3242, 10595}, {3311, 12257}, {3312, 12256}, {3313, 7383}, {3416, 5818}, {3424, 14492}, {3448, 9970}, {3522, 5092}, {3529, 6329}, {3541, 3567}, {3542, 6403}, {3543, 11179}, {3546, 5462}, {3547, 5446}, {3549, 12363}, {3620, 5056}, {3629, 3855}, {3763, 5067}, {3767, 5052}, {3818, 3832}, {3851, 11008}, {4259, 6833}, {4260, 6847}, {4265, 6942}, {5012, 7500}, {5026, 13172}, {5064, 11245}, {5095, 7687}, {5096, 6950}, {5133, 6515}, {5135, 6934}, {5306, 9756}, {5486, 14483}, {5510, 10761}, {5511, 10760}, {5512, 10765}, {5587, 5847}, {5654, 11188}, {5820, 6968}, {6034, 14651}, {6193, 7528}, {6225, 11403}, {6243, 14786}, {6248, 6392}, {6337, 10983}, {6593, 12383}, {6623, 8541}, {6756, 11426}, {6846, 10477}, {7000, 7585}, {7374, 7586}, {7386, 10601}, {7395, 11821}, {7398, 9306}, {7400, 11574}, {7403, 11411}, {7408, 13366}, {7409, 11550}, {7493, 14389}, {7519, 11003}, {7533, 11004}, {7694, 7753}, {7709, 13331}, {7735, 13860}, {7772, 8721}, {7774, 13862}, {7803, 13354}, {8540, 10590}, {8593, 9880}, {8889, 13567}, {9147, 14397}, {9732, 11291}, {9733, 11292}, {9744, 9993}, {9749, 14136}, {9750, 14137}, {9815, 13346}, {9935, 12242}, {10151, 11405}, {10201, 13451}, {10591, 12589}, {10596, 12594}, {10597, 12595}, {10598, 12586}, {10599, 12587}, {11160, 11178}, {11425, 11745}

X(14853) = midpoint of X(i) and X(j) for these {i,j}: {381, 5093}, {3839, 5032}, {5102, 10516}
X(14853) = reflection of X(i) in X(j) for these {i,j}: {2, 14561}, {376, 5085}, {1992, 5093}, {5085, 597}, {7709, 13331}, {10519, 2}, {14561, 5476}, {14651, 6034}
X(14853) = crosspoint of X(4) and X(14494)
X(14853) = crossdifference of every pair of points on line {520, 3288}
X(14853) = crosssum of X(3) and X(5050)
X(14853) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 9748, 9753), (2, 9753, 9752), (4, 6, 6776), (5, 1351, 69), (6, 5480, 4), (193, 3091, 1352), (262, 9748, 9752), (262, 9753, 2), (381, 1992, 11180), (427, 9777, 11433), (428, 11402, 11206), (576, 1352, 193), (1350, 3589, 631), (1587, 6202, 4), (1588, 6201, 4), (3087, 10002, 4), (3832, 5921, 3818), (6403, 9781, 9969)
X(14853) = orthosymmedial-circle-inverse of X(6776)
X(14853) = X(14495)-anticomplementary conjugate of X(8)

X(14854) =  (name pending)

Barycentrics    (a^2 - b^2 - c^2)*(a^2 - a*b + b^2 - c^2)*(a^2 + a*b + b^2 - c^2)*(a^2 - b^2 - a*c + c^2)*(a^2 - b^2 + a*c + c^2)*(a^12 - 3*a^10*b^2 + 4*a^8*b^4 - 6*a^6*b^6 + 9*a^4*b^8 - 7*a^2*b^10 + 2*b^12 - 3*a^10*c^2 + 7*a^8*b^2*c^2 - 2*a^6*b^4*c^2 - 11*a^4*b^6*c^2 + 15*a^2*b^8*c^2 - 6*b^10*c^2 + 4*a^8*c^4 - 2*a^6*b^2*c^4 + 10*a^4*b^4*c^4 - 8*a^2*b^6*c^4 + 6*b^8*c^4 - 6*a^6*c^6 - 11*a^4*b^2*c^6 - 8*a^2*b^4*c^6 - 4*b^6*c^6 + 9*a^4*c^8 + 15*a^2*b^2*c^8 + 6*b^4*c^8 - 7*a^2*c^10 - 6*b^2*c^10 + 2*c^12) : :
X(14854) = 2 X(265) + X(5961) = 4 X(125) - X(13496)

X(14854) lies on the Lester circle and this line: {3, 125}

X(14855) =  X(3)X(64)∩X(20)X(52)

Barycentrics    a^2*(a^6*b^2 - 3*a^4*b^4 + 3*a^2*b^6 - b^8 + a^6*c^2 + 10*a^4*b^2*c^2 - 9*a^2*b^4*c^2 - 2*b^6*c^2 - 3*a^4*c^4 - 9*a^2*b^2*c^4 + 6*b^4*c^4 + 3*a^2*c^6 - 2*b^2*c^6 - c^8) : :
X(14855) = 2 X(20) + X(52) = X(185) + 2 X(550) = 2 X(389) + X(1657) = 3 X(376) - X(2979) = 2 X(1216) - 5 X(3522) = 3 X(3) - 2 X(3819) = 3 X(373) - 2 X(3845) = 5 X(3567) + X(5059) = X(3529) + 2 X(5446) = 7 X(3528) - 4 X(5447) = X(3146) - 4 X(5462) = 4 X(548) - X(5562) = 4 X(3819) - 3 X(5891) = 5 X(5891) - 4 X(5907) = 5 X(3819) - 3 X(5907) = 5 X(3) - 2 X(5907) = 5 X(51) - 6 X(5946) = 2 X(1216) + X(6241) = 5 X(3522) + X(6241) = 3 X(381) - 4 X(6688) = X(382) - 4 X(9729) = 4 X(5946) - 5 X(9730) = 2 X(51) - 3 X(9730) = X(5073) - 4 X(10110) = 3 X(3524) - 2 X(10170) = 2 X(5446) - 5 X(10574) = X(3529) + 5 X(10574) = 2 X(3) + X(10575) = 4 X(3819) + 3 X(10575)

X(14855) lies on these lines: {2, 11455}, {3, 64}, {4, 5892}, {20, 52}, {22, 8717}, {30, 51}, {74, 6636}, {113, 1368}, {140, 11381}, {184, 10564}, {185, 550}, {373, 3845}, {376, 2979}, {381, 6688}, {382, 9729}, {389, 1657}, {511, 3534}, {512, 5664}, {548, 5562}, {569, 12085}, {631, 12279}, {1060, 11189}, {1209, 6247}, {1216, 3522}, {1370, 4846}, {1593, 13336}, {1656, 13474}, {2781, 9967}, {3060, 11001}, {3146, 5462}, {3523, 12290}, {3524, 10170}, {3525, 11439}, {3528, 5447}, {3529, 5446}, {3567, 5059}, {3627, 13364}, {3830, 5943}, {3843, 11695}, {3917, 5663}, {4550, 7485}, {5012, 7464}, {5020, 11820}, {5073, 10110}, {5650, 12100}, {6102, 12103}, {6243, 13382}, {7391, 7706}, {7484, 11472}, {7689, 10323}, {10304, 11459}, {10620, 14810}, {10984, 12084}, {11402, 13352}, {11438, 12083}, {12099, 12295}, {12163, 12166}

X(14855) = midpoint of X(i) and X(j) for these {i,j}: {20, 5890}, {3060, 11001}, {5891, 10575}, {5892, 14641}
X(14855) = reflection of X(i) in X(j) for these {i,j}: {4, 5892}, {52, 5890}, {3627, 13364}, {3830, 5943}, {3917, 8703}, {5891, 3}, {12162, 5891}, {12295, 12099}
X(14855) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 10575, 12162), (185, 550, 10625), (548, 13491, 5562), (3522, 6241, 1216), (3528, 12111, 5447), (3529, 10574, 5446)

X(14856) =  X(4)X(542)∩X(1499)X(2686)

Barycentrics    (SB-SC)^2*(3*SA-2*SW)*(SA^2+2* SB*SC) : :

See Antreas Hatzipolakis and César Lozada, Hyacinthos 26662.

X(14856) lies on these lines: {4, 542}, {1499, 2686}, {1513, 1533}

X(14856) = reflection of X(6791) in X(5512)
X(14856) = orthopole of line X(2)X(6)

X(14857) =  X(30)X(113)∩X(125)X(13152)

Barycentrics    (SB-SC)^2*(S^2-3*SB*SC)*(3*R^2-2*SA)*(SA^2-3*S^2) : :

See Antreas Hatzipolakis and César Lozada, Hyacinthos 26662.

X(14857) lies on these lines: {30, 113}, {125, 13152}, {137, 1510}

X(14858) =  X(2)X(6)∩X(1499)X(2686)

Barycentrics    (SB-SC)^2*(3*SA-2*SW)*(6*(36* R^2-5*SW)*S^2-2*(9*SA^2-6*SA* SW+4*SW^2)*SW) : :

See Antreas Hatzipolakis and César Lozada, Hyacinthos 26662.

X(14858) lies on these lines: {2, 6}, {1499, 2686}

X(14859) =  CEVAPOINT OF X(51) AND X(1989)

Barycentrics    b^2 c^2 (a^2-a b+b^2-c^2)^2 (a^2+a b+b^2-c^2)^2 (-a^2+b^2-a c-c^2)^2 (-a^2+b^2+a c-c^2)^2 (a^4-2 a^2 b^2+b^4-a^2 c^2-b^2 c^2) (-a^4+a^2 b^2+2 a^2 c^2+b^2 c^2-c^4) : :

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 26664.

X(14859) lies on this line: {476,1141}

X(14859) = cevapoint of X(51) and X(1989)
X(14859) = X(51)-cross conjugate of X(1989)
X(14859) = X(i)-isoconjugate of X(j) for these (i,j): {323, 2290}, {1154, 6149}
X(14859) = barycentric product X(i)*X(j) for these {i,j}: {94, 1141}, {276, 14595}
X(14859) = barycentric quotient X(i)/X(j) for these {i,j}: {94, 1273}, {1141, 323}, {1989, 1154}, {8882, 3043}, {14595, 216}

X(14860) =  ISOGONAL CONJUGATE OF X(13367)

Barycentrics    SB*SC *(SB+8*R^2-3*SW)*(SC+8*R^2-3*SW) : :

See Antreas Hatzipolakis and César Lozada, Hyacinthos 26669.

The trilinear polar of X(14860) passes through X(2501). (Randy Hutson, November 2, 2017)

X(14860) lies on these lines:

{5, 8884}, {225, 7541}, {264, 7507}, {381, 1093}, {393, 3091}, {403, 1179}, {427, 1105}, {648, 3574}, {847, 7547}, {1300, 1594}, {1826, 7563}, {3832, 6526}, {6531, 7745}, {7566, 14249}

X(14860) = cevapoint of X(4) and X(5)

X(14861) =  X(54)X(550)∩X(74)X(140)

Barycentrics    SA*(3*S^2+7*SB^2)*(3*S^2+7*SC^2) : :
X(14861) = 7*X(4)-12*X(13566)

See Antreas Hatzipolakis and César Lozada, Hyacinthos 26669.

X(14861) lies on the Jerabek hyperbola and these lines: {2, 13452}, {4, 12006}, {6, 1657}, {20, 13472}, {30, 1173}, {54, 550}, {64, 1656}, {65, 4857}, {74, 140}, {185, 3519}, {3426, 3851}, {3431, 3522}, {3523, 11270}, {3527, 5073}, {3858, 13603}, {5056, 11738}, {5663, 5900}, {6697, 13093}

X(14862) =  X(4)X(54)∩X(140)X(6000)

Barycentrics    3*(4*R^2+SA-2*SW)*S^2+(28*R^2-SW)*SB*SC : :
X(14862) = 5*X(4)+3*X(9833) = 7*X(4)+9*X(11206) = X(64)-3*X(10193) = 5*X(140)-3*X(6696) = 9*X(154)-X(1657) = X(550)+3*X(2883) = X(550)-3*X(10282) = 3*X(1498)+5*X(1656) = 7*X(9833)-15*X(11206)

See Antreas Hatzipolakis and César Lozada, Hyacinthos 26669.

X(14862) lies on these lines: {4, 54}, {18, 10675}, {64, 10193}, {140, 6000}, {154, 1657}, {399, 3519}, {550, 1511}, {1498, 1656}, {1503, 3850}, {2393, 12002}, {3357, 3523}, {3522, 5878}, {5056, 14216}, {5073, 14530}, {5270, 10535}, {5562, 6053}, {5655, 13564}, {5972, 10575}, {6225, 10299}, {6747, 6750}, {8960, 12970}, {10112, 11799}, {10116, 11563}

X(14862) = midpoint of X(i) and X(j) for these {i,j}: {2883, 10282}, {5656, 10182}
X(14862) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4, 1614, 10619), (4, 10619, 13403)

X(14863) =  X(20)X(2889)∩X(1249)X(3533)

Barycentrics    (3*S^4-(SB+6*SC)*SA*S^2+(SB+SC)*SA^2*SB)*(3*S^4-(SC+6*SB)*SA*S^2+(SC+SB)*SA^2*SC) : :

See Antreas Hatzipolakis and César Lozada, Hyacinthos 26669.

X(14863) lies on these lines: {20, 2889}, {1249, 3533}, {3851, 14249}

X(14864) =  X(4)X(51)∩X(140)X(1503)

Barycentrics    3*(4*R^2-SA)*S^2-(4*R^2+5*SW)*SB*SC : :
X(14864) = 7*X(4)-3*X(5878) = 11*X(4)-3*X(6225) = 5*X(4)+3*X(12324) = X(4)+3*X(14216) = 3*X(64)+X(5073) = 10*X(140)-9*X(10182) = 4*X(140)-3*X(10282) = 11*X(5878)-7*X(6225) = 5*X(5878)+7*X(12324) = X(5878)+7*X(14216) = 5*X(6225)+11*X(12324) = 6*X(10182)-5*X(10282)

See Antreas Hatzipolakis and César Lozada, Hyacinthos 26669.

X(14864) lies on these lines: {4, 51}, {26, 11645}, {64, 5073}, {140, 1503}, {427, 12242}, {550, 6247}, {1498, 3851}, {1656, 1853}, {1657, 3357}, {2781, 13421}, {2883, 3858}, {3519, 10625}, {3523, 9833}, {3533, 11206}, {3818, 11695}, {3854, 5656}, {6240, 13399}, {6288, 14855}, {10619, 11430}

X(14864) = {X(11457), X(11550)}-harmonic conjugate of X(389)

X(14865) =  EULER LINE INTERCEPT OF X(74)X(389)

Trilinears    4 cos A + 3 sec A : :
Barycentrics    SB*SC*(SB+SC) *(7*SA^2+3*S^2) : :
X(14865) = 4*X(3)-3*X(7512) = 5*X(3)-3*X(13564) = X(3)-3*X(14130)

As a point of the Euler line, X(14865) has Shinagawa coefficients (4F, 3E - 4F).

See Antreas Hatzipolakis and César Lozada, Hyacinthos 26677.

X(14865) lies on these lines: {2, 3}, {6, 13452}, {39, 8744}, {52, 11440}, {54, 6000}, {61, 11476}, {62, 11475}, {64, 7592}, {74, 389}, {93, 1105}, {112, 5007}, {184, 12290}, {185, 1199}, {323, 5876}, {511, 6242}, {567, 13491}, {575, 12294}, {578, 6241}, {1204, 3567}, {1493, 2914}, {1503, 12254}, {1614, 11381}, {1830, 5563}, {1870, 3746}, {1968, 7772}, {1970, 13509}, {2777, 3574}, {2935, 6696}, {3043, 5609}, {3060, 7689}, {3092, 6426}, {3093, 6425}, {3357, 5890}, {3431, 9707}, {3448, 12370}, {4550, 11444}, {5012, 10575}, {5206, 10986}, {5237, 10633}, {5238, 10632}, {5352, 10641}, {5410, 6447}, {5411, 6448}, {5866, 7752}, {5921, 9925}, {6152, 13391}, {6153, 7691}, {6247, 12022}, {6419, 11474}, {6420, 11473}, {6453, 10880}, {6454, 10881}, {6530, 14634}, {6748, 11063}, {6759, 11455}, {7722, 11536}, {8567, 11270}, {9781, 11438}, {9820, 12302}, {10095, 12041}, {10539, 11439}, {10606, 10982}, {10620, 14627}, {11402, 13093}, {11425, 11456}, {11441, 11472}, {11459, 13346}, {11550, 12289}, {12111, 12364}, {12134, 12383}, {13293, 14644}, {13367, 13474}, {13434, 13445}

X(14865) = isogonal conjugate of X(14861)
X(14865) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 4, 3518), (3, 1597, 5198), (3, 3146, 12088), (3, 3627, 23), (4, 1593, 13596), (24, 1597, 4), (378, 1593, 4), (378, 13596, 186), (1593, 3516, 1597), (1594, 1885, 4), (1595, 6240, 4), (1597, 3516, 24), (3520, 13596, 4), (5073, 7502, 12087)

X(14866) =  MIDPOINT OF X(4) AND X(12505)

Barycentrics    4*a^10-7*(b^2+c^2)*a^8-15*(b^ 2-c^2)^2*a^6+(b^2+c^2)*(13*b^ 4-40*b^2*c^2+13*c^4)*a^4+(11* b^8+11*c^8-2*b^2*c^2*(14*b^4- 9*b^2*c^2+14*c^4))*a^2+(-c^4+ b^4)*(b^2-c^2)*(-6*b^4+18*b^2* c^2-6*c^4) : :
X(14866) = 2*X(3)-3*X(10163) = 4*X(5)-3*X(10162) = X(20)-3*X(9829) = 7*X(3090)-6*X(10173) = 5*X(3091)-3*X(6032) = X(3146)+3*X(6031) = 3*X(10162)-2*X(12506)

See Thanos Kalogerakis and César Lozada, Hyacinthos 266778.

X(14866) lies on these lines: {3, 10163}, {4, 3849}, {5, 9172}, {20, 9829}, {3090, 10173}, {3091, 6032}, {3146, 6031}, {7527, 14682}

X(14866) = midpoint of X(4) and X(12505)
X(14866) = reflection of X(12506) in X(5)
X(14866) = complement of X(34792)
X(14866) = anticomplement of X(31762)
X(14866) = X(20)-of-5th-Euler-triangle
X(14866) = X(12506)-of-Johnson-triangle
X(14866) = {X(5), X(12506)}-harmonic conjugate of X(10162)

X(14867) =  X(3)X(10166)∩X(20)X(353)

Barycentrics    5*(b^2+c^2)*a^8-(20*b^4-b^2*c^ 2+20*c^4)*a^6+3*(b^2+c^2)*(5* b^4-11*b^2*c^2+5*c^4)*a^4+(2* b^8+2*c^8-b^2*c^2*(7*b^4-18*b^ 2*c^2+7*c^4))*a^2-(b^4-c^4)*( b^2-c^2)*(2*b^2-c^2)*(b^2-2*c^ 2) : :
X(14867) = 2*X(3)-3*X(10166) = X(20)-3*X(353) = 5*X(631)-6*X(10160)

See Thanos Kalogerakis and César Lozada, Hyacinthos 266778.

X(14867) lies on these lines: {3, 10166}, {4, 9830}, {20, 353}, {524, 12505}, {631, 10160}, {6233, 12110}

X(14867) = complement of X(34795)

X(14868) =  X(21)X(90)∩X(28)X(662)

Barycentrics    a*(a^2+2*(b+c)*a-(b+c)^2)*(-a^ 2+b^2+c^2)*(a+b)*(a+c) : :
See Antreas Hatzipolakis and César Lozada, Hyacinthos 26682.

X(14868) lies on these lines: {1, 1958}, {3, 1812}, {21, 90}, {28, 662}, {72, 1444}, {73, 7364}, {78, 1790}, {81, 386}, {86, 443}, {100, 3193}, {283, 4855}, {333, 631}, {572, 2287}, {1437, 1792}, {1959, 2939}, {2327, 10884}, {4197, 5333}, {4658, 8897}

X(14868) = {X(1437), X(5440)}-harmonic conjugate of X(1792)

X(14869) =  17th HATZIPOLAKIS-MOSES-EULER POINT

Barycentrics    8 a^4-11 a^2 b^2+3 b^4-11 a^2 c^2-6 b^2 c^2+3 c^4 : :
X(14869) = 9 X(2) + 5 X(3), 11 X(3) + 3 X(4) = 4 X(4) - 11 X(5) = 12 X(2) - 5 X(5) = 4 X(3) + 3 X(5) = 17 X(3) - 3 X(20) = 17 X(5) + 4 X(20) = 17 X(4) + 11 X(20) = 3 X(2) - 10 X(140) = X(5) - 8 X(140) = X(3) + 6 X(140) = 19 X(5) - 12 X(381) = 19 X(2) - 5 X(381) = 19 X(3) + 9 X(381) = 15 X(2) - X(382) = 3 X(382) - 10 X(546) = 15 X(5) - 8 X(546) = 9 X(2) - 2 X(546) = 15 X(140) - X(546) = 5 X(3) + 2 X(546) = 17 X(2) - 10 X(547) = 17 X(140) - 3 X(547) = X(20) + 6 X(547) = 17 X(3) + 18 X(547) = 13 X(3) - 6 X(548) = 13 X(140) + X(548) = 13 X(5) + 8 X(548) = 13 X(546) + 15 X(548) = 2 X(3) - 9 X(549) = 4 X(140) + 3 X(549) = 2 X(2) + 5 X(549) = X(5) + 6 X(549) = 4 X(547) + 17 X(549) = 2 X(381) + 19 X(549) = 10 X(20) - 17 X(550) = 10 X(3) - 3 X(550) = 15 X(549) - X(550) = 6 X(2) + X(550) = 5 X(5) + 2 X(550) = 4 X(546) + 3 X(550) = 2 X(382) + 5 X(550) = 10 X(4) + 11 X(550) = X(3) - 15 X(631) = 3 X(549) - 10 X(631) = 2 X(140) + 5 X(631) = 3 X(5) - 10 X(632) = 12 X(140) - 5 X(632) = 6 X(631) + X(632) = 2 X(3) + 5 X(632) = 9 X(549) + 5 X(632) = 13 X(632) - 6 X(1656) = 13 X(631) + X(1656) = 2 X(548) + 5 X(1656) = 13 X(3) + 15 X(1656) = 9 X(381) - 19 X(3090) = 18 X(547) - 17 X(3090) = 15 X(1656) - 13 X(3090) = 3 X(4) - 11 X(3090) = 9 X(2) - 5 X(3090)

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 26685.

X(14869) lies on these lines: {2,3}, {182,3631}, {499,10386}, {575,3629}, {576,6329}, {590,6454}, {615,6453}, {1385,3626}, {1483,3632}, {1484,6154}, {1587,6522 }, {1588,6519}, {2883,10193}, {3055,5206}, {3244,5690}, {3564, 10541}, {3592,5420}, {3594,5418} ,{3636,6684}, {3746,5433}, {3819,6102}, {3917,12006}, {4031, 6147}, {5010,7294}, {5050,11008} ,{5092,5944}, {5326,7280}, {5432,5563}, {5447,5946}, {5609, 6699}, {5650,11591}, {5892,6101} ,{5901,7991}, {6221,13993}, {6247,10182}, {6398,13925}, {6407 ,13939}, {6408,13886}, {6419, 13966}, {6420,8981}, {6425,7584}, {6426,7583}, {6427,9540}, {6428,13935}, {7843,9771}, {7863, 11168}, {7982,10283}, {9143, 13393}, {9680,13847}, {10170, 13491}, {10263,11695}, {10519, 11482}, {10625,11592}, {13372, 14073}

X(14869) = midpoint of X(i) and X(j) for these {i,j}: {2,15700}, {3, 3090}, {3523, 3526}, {3528, 3851}
X(14869) = reflection of X(i) in X(j) for these {i,j}: {550, 3528}, {3526, 140}, {3857, 3090}
X(14869) = complement X(3851)
X(14869) = X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 3, 546), (2, 550, 5), (2, 3523, 3528), (2, 3528, 3851), (2, 3529, 5079), (2, 3530, 550), (2, 3855, 1656), (2, 5079, 3628), (2, 10299, 382), (2, 14269, 547), (3, 140, 632), (3, 546, 550), (3, 631, 12108), (3, 632, 5), (3, 1656, 3146), (3, 3091, 12103), (3, 3146, 548), (3, 3525, 3628), (3, 3526, 3090), (3, 3628, 3627), (3, 5054, 10303), (3, 5070, 5076), (3, 5072, 20), (3, 5076, 376), (3, 5079, 3529), (3, 10303, 140), (3, 12108, 549), (5, 140, 11539), (20, 547, 3858), (20, 5072, 12102), (140, 548, 10124), (140, 549, 5), (140, 631, 549), (140, 3530, 2), (140, 3628, 3525), (140, 11812, 631), (140, 12108, 3), (376, 5070, 3850), (546, 3529, 3627), (546, 3530, 3), (546, 3628, 5079), (546, 3851, 3857), (546, 12102, 14269), (546, 12811, 3855), (546, 12812, 11737), (547, 3858, 5), (547, 12102, 5072), (548, 1656, 3845), (548, 10124, 1656), (548, 12811, 3146), (549, 550, 3530), (549, 632, 3), (549, 3845, 3524), (549, 11539, 8703), (550, 3627, 3529), (631, 5054, 140), (631, 10303, 3), (632, 3627, 3628), (1656, 3146, 12811), (1656, 3524, 548), (1656, 3845, 5), (1657, 5067, 5066), (3090, 3523, 3), (3090, 3857, 5), (3091, 12103, 3627), (3146, 3524, 3), (3146, 12811, 3845), (3522, 5055, 3853), (3524, 10124, 3845), (3525, 3529, 2), (3525, 3628, 632), (3526, 3851, 2), (3529, 5079, 546), (3534, 5056, 3861), (3627, 3628, 5), (3628, 12103, 3091), (5010, 7294, 10593), (5054, 11812, 549), (5071, 5073, 3856), (5072, 12102, 3858), (5326, 7280, 10592), (10303, 12108, 632), (11592, 13363, 10625), (14784, 14785, 3854)

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Simpedal points: X(14870)-X(14889)

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This preamble and centers X(14870)-X(14889) were contributed by César Eliud Lozada, October 19, 2017.

"To determine a point P whose pedal triangle A'B'C' with regard to a given triangle ABC shall be similar to a given triangle A*B*C*". (Johnson, Roger A.: Advanced Eucliden Geometry, Dover, New York, 1960, problem 205, pp. 142.)

Assume ABC and A*B*C* are not inversely similar. For finding P, take any two points B1, C1 on AC and AB, respectively, and build A1B1C1 directly similar to A*B*C*. Then, through A2=AA1∩BC, draw parallels to A1B1 and A1C1, cutting AC and AB at B2 and C2, respectively. The required P is the Miquel point of A2B2C2 with respect to ABC. (Proof in the cited reference).

The point P and the triangle A'B'C' will be named here the simpedal point of A*B*C* in ABC and the simpedal triangle of A*B*C* in ABC, respectively.

Let A* = UA : VA : WA (trilinears) and similarly B* and C*. Let M be the trilinear matrix of A*B*C* and mij the (i, j)-minor of M. Denote δij = (-1)i+j mij. Then,

 P = a (b δ33 δ21 (a^2 - b^2) + c δ22 δ31 (a^2 - c^2) + b c (b δ32 δ21 + c δ23 δ31) - (δ21 δ31 + δ22 δ32 + δ23 δ33) a b c + a δ22 δ33 (b^2 + c^2 - a^2))/(a UA + b VA + c WA) : :

The appearance of (T, I, J) in the following list means that X(I) is the simpedal point of triangle T in ABC and X(J) is the simpedal point of ABC in triangle T:
(anticomplementary, 3, 4), (2nd Brocard, 23, 110), (3rd Brocard, 14870, 14871), (4th Brocard, 187, 115), (5th Brocard, 3, 9821), (circummedial, 2, 2), (circumorthic, 4, 4), (1st circumperp, 1, 40), (2nd circumperp, 1, 1), (circumsymmedial, 6, 6), (Euler, 3, 5), (excentral, 1, 1), (extangents, 4, 40), (extouch, 40, 14872), (Feuerbach, 501, 14873), (Fuhrmann, 36, 80), (inner-Grebe, 3, 1161), (outer-Grebe, 3, 1160), (hexyl, 1, 40), {incentral, 501, (incentral, 501, 14874), (intangents, 4, 1), (intouch, 1, 65), (Johnson, 3, 4), (Kosnita, 4, 3), (Lemoine, 14875, 14876), (Macbeath, 6801, 14877), (medial, 3, 5), (midheight, 20, 185), (mixtilinear, 14878, 14879), (inner-Napoleon, 16, 14), (outer-Napoleon, 15, 13), (1st Neuberg, 32, 14880), (2nd Neuberg, 39, 14881), (orthic, 4, 4), (reflection, 5, 10263), (1st Sharygin, 1, 2292), (2nd Sharygin, 1, 2254), (1st Schiffler, 14882, 14883), (2nd Schiffler, 56, 12740), (Steiner, 14366, 14884), (submedial, 4, 5), (symmedial, 14885, 14886), (tangential, 3, 4), (Trinh, 4, 3), (Yff contact, 14887, 14888), (inner-Yff, 3, 55), (outer-Yff, 3, 56), (Yiu, 5961, 14889)

X(14870) = SIMPEDAL POINT OF 3rd BROCARD TRIANGLE IN ABC

Barycentrics    ((a^2+b^2)^2-a^2*b^2)*(-a^2*c^2+(a^2+c^2)^2)*(b^4*c^4*a^10-(b^6+c^6)*(b^2+c^2)^2*a^8+(b^12+c^12+(b^8+c^8+4*(b^4+b^2*c^2+c^4)*b^2*c^2)*b^2*c^2)*a^6-b^4*c^4*(b^2+c^2)*(2*b^4+b^2*c^2+2*c^4)*a^4-b^6*c^6*(b^4-b^2*c^2+c^4)*a^2+b^8*c^8*(b^2+c^2))*a^2 : :

X(14870) lies on the line {1691,9467}

X(14871) = SIMPEDAL POINT OF ABC IN 3rd BROCARD TRIANGLE

Barycentrics    b^4*c^4*a^18-(b^2+c^2)*(b^8+c^8+b^2*c^2*(b^4-b^2*c^2+c^4))*a^16+(b^12+c^12-(b^8+c^8-b^2*c^2*(2*b^4+3*b^2*c^2+2*c^4))*b^2*c^2)*a^14+(b^2+c^2)*(2*b^8+2*c^8-3*b^2*c^2*(b^4-b^2*c^2+c^4))*b^2*c^2*a^12+(b^2-c^2)^2*b^4*c^4*(3*b^4+2*b^2*c^2+3*c^4)*a^10-(b^6+c^6)*(b^2+c^2)^2*b^4*c^4*a^8-(3*b^8+3*c^8-b^2*c^2*(2*b^4+b^2*c^2+2*c^4))*b^6*c^6*a^6+(b^4+b^2*c^2+c^4)*(b^2+c^2)*b^8*c^8*a^4+b^8*c^8*(b^8-b^4*c^4+c^8)*a^2-b^12*c^12*(b^2+c^2) : :

X(14871) lies on the line {385,706}

X(14872) = SIMPEDAL POINT OF ABC IN EXTOUCH TRIANGLE

Barycentrics    a*((b+c)*a^5-(b-c)^2*a^4-2*(b+c)*(b^2+c^2)*a^3+2*(b^2+c^2)^2*a^2+(b^2-c^2)^2*(b+c)*a-(b^2-c^2)^2*(b+c)^2) : :
X(14872) = 2*X(3)-3*X(210) = 4*X(5)-3*X(354) = X(20)-3*X(3681) = 3*X(210)-X(12680) = 3*X(392)-2*X(5882) = 5*X(631)-6*X(3740) = 2*X(942)-3*X(5587) = 2*X(946)-3*X(5927) = X(3555)-3*X(5927) = 3*X(5587)-4*X(9947)

X(14872) lies on these lines: {1,1864}, {2,12675}, {3,210}, {4,518}, {5,354}, {8,6001}, {20,3681}, {30,7957}, {35,5531}, {40,971}, {44,602}, {55,5534}, {56,5720}, {63,11500}, {65,68}, {72,515}, {78,12114}, {84,200}, {90,11508}, {101,947}, {140,12757}, {144,12670}, {153,2894}, {219,1903}, {382,517}, {392,5882}, {480,3358}, {497,5811}, {519,12672}, {631,3740}, {942,5290}, {944,960}, {946,3555}, {952,1898}, {956,6261}, {962,9804}, {999,9850}, {1006,5302}, {1012,3811}, {1155,11499}, {1158,5687}, {1385,5251}, {1456,8757}, {1490,3428}, {1519,3813}, {1532,10916}, {1617,1728}, {1656,13373}, {1698,9940}, {1709,10306}, {1753,12136}, {1765,3694}, {1837,10629}, {1858,5252}, {2182,9798}, {2551,5768}, {2771,12751}, {2800,10914}, {2810,5907}, {3072,4641}, {3073,3744}, {3085,10391}, {3090,3742}, {3091,3873}, {3146,4661}, {3295,5779}, {3338,6918}, {3419,6256}, {3475,6846}, {3576,5044}, {3579,5918}, {3633,13600}, {3678,4297}, {3683,10267}, {3689,11248}, {3697,6684}, {3698,5790}, {3711,7171}, {3715,8273}, {3751,5706}, {3753,5884}, {3812,5818}, {3817,3881}, {3832,4430}, {3848,5067}, {3868,7686}, {3870,11496}, {3876,5731}, {3916,6796}, {3929,10268}, {3953,5400}, {3983,13369}, {4015,10164}, {4326,7160}, {4420,6909}, {4640,11491}, {4662,5657}, {4847,6260}, {4853,7971}, {5045,8227}, {5049,9624}, {5173,9612}, {5227,5776}, {5439,10175}, {5440,5450}, {5562,8679}, {5570,10826}, {5572,5817}, {5686,12669}, {5721,13161}, {5781,7719}, {5794,12115}, {5815,9799}, {6244,12684}, {6282,9954}, {6705,6745}, {6765,12705}, {6766,8001}, {6826,10404}, {6850,12678}, {7080,14647}, {7082,11510}, {7982,9856}, {9037,11412}, {9052,13598}, {9848,9957}, {9956,10202}, {10394,12710}, {10698,11256}, {10827,13750}, {10861,11024}, {11414,12329}, {12608,13257}

X(14872) = midpoint of X(i) and X(j) for these {i,j}: {8, 12528}, {5691, 5904}, {5693, 5881}
X(14872) = reflection of X(i) in X(j) for these (i,j): (1, 5777), (65, 355), (942, 9947), (944, 960), (3057, 5887), (3555, 946), (3633, 13600), (3868, 7686), (3893, 12645), (4297, 3678), (6282, 9954), (7982, 9856), (9943, 4662), (11827, 12527), (12671, 11500), (12680, 3), (14100, 5779), (14110, 72)
X(14872) = anticomplement of X(12675)
X(14872) = extouch-isogonal conjugate of X(40)
X(14872) = Mandart-circle-inverse-of-X(13528)
X(14872) = { X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (84, 200, 10310), (210, 12680, 3), (942, 9947, 5587), (1158, 5687, 13528), (3091, 3873, 13374), (3555, 5927, 946), (3697, 10167, 6684), (4662, 9943, 5657), (4882, 7992, 40), (5534, 7330, 55), (6737, 12059, 72), (12528, 12529, 12666)

X(14873) = SIMPEDAL POINT OF ABC IN FEUERBACH TRIANGLE

Barycentrics    (b+c)*((b+c)*a^5+2*(b^2+b*c+c^2)*a^4+(b+c)*(b^2+b*c+c^2)*a^3-(b^4+c^4-b*c*(b+c)^2)*a^2-(b^2-c^2)*(b-c)*(2*b^2+3*b*c+2*c^2)*a-(b+c)*(b^2-c^2)*(b^3-c^3)) : :
X(14873) = X(1)-3*X(14844)

X(14873) lies on these lines: {1,14844}, {4,14192}, {5,580}, {12,502}, {125,12047}, {429,1785}, {442,1155}, {495,13514}, {942,8287}, {1834,12019}, {2653,6627}, {3142,8068}, {5949,8728}, {6841,7687}, {8286,9955}

X(14873) = polar circle-inverse-of-X(14192)

X(14874) = SIMPEDAL POINT OF ABC IN INCENTRAL TRIANGLE

Barycentrics    a*(a^6+(b+c)*a^5-(b+c)^2*a^4-(b+c)*(2*b^2+b*c+2*c^2)*a^3-(b^4+c^4+b*c*(b^2+b*c+c^2))*a^2+(b^2-c^2)^2*(b+c)*a+(b^2-c^2)^2*(b+c)^2) : :

X(14874) lies on these lines: {1,5180}, {42,502}, {58,7073}, {1834,12019}, {3743,5184}, {4336,13514}, {5496,5497}

X(14875) = SIMPEDAL POINT OF LEMOINE TRIANGLE IN ABC

Barycentrics    (a^6+(b^2+c^2)*a^4-(b^4+17*b^2*c^2+c^4)*a^2-(b^2+c^2)*(b^4+8*b^2*c^2+c^4))*(2*a^2-b^2+2*c^2)*(a^2+4*b^2+c^2)*(2*a^2+2*b^2-c^2)*(a^2+b^2+4*c^2)*a^2 : :

X(14875) lies on these lines: {}

X(14876) = SIMPEDAL POINT OF ABC IN LEMOINE TRIANGLE

Barycentrics    (3*a^12+13*(b^2+c^2)*a^10-2*(3*b^4-8*b^2*c^2+3*c^4)*a^8-(b^2+c^2)*(28*b^4+151*b^2*c^2+28*c^4)*a^6+(b^8+c^8-b^2*c^2*(170*b^4+477*b^2*c^2+170*c^4))*a^4+(b^2+c^2)*(15*b^8+15*c^8+b^2*c^2*(b^4-82*b^2*c^2+c^4))*a^2+2*(b^4-c^4)^2*(b^2+c^2)^2)*(2*a^2-b^2+2*c^2)*(2*a^2+2*b^2-c^2) : :

X(14876) lies on these lines: {}

X(14877) = SIMPEDAL POINT OF ABC IN MACBEATH TRIANGLE

Barycentrics    (a^2+b^2-c^2)*(a^2-b^2+c^2)*(a^18-5*(b^2+c^2)*a^16+2*(5*b^4+8*b^2*c^2+5*c^4)*a^14-5*(b^2+c^2)*(2*b^4+b^2*c^2+2*c^4)*a^12+(4*b^8+4*c^8-b^2*c^2*(2*b^4+b^2*c^2+2*c^4))*a^10+(b^2+c^2)*(4*b^8+4*c^8+b^2*c^2*(3*b^4+4*b^2*c^2+3*c^4))*a^8-2*(b^2-c^2)^2*(5*b^8+5*c^8+2*b^2*c^2*(2*b^4+3*b^2*c^2+2*c^4))*a^6+(b^4-c^4)*(b^2-c^2)*(10*b^8+10*c^8-b^2*c^2*(19*b^4-16*b^2*c^2+19*c^4))*a^4-(b^2-c^2)^4*(5*b^8+5*c^8-b^2*c^2*(2*b^4+3*b^2*c^2+2*c^4))*a^2+(b^6+c^6)*(b^2-c^2)^6) : :

X(14877) lies on these lines: {4,13418}, {324,6798}

X(14878) = SIMPEDAL POINT OF MIXTILINEAR TRIANGLE IN ABC

Barycentrics    a^2*(a+b-c)*(a-b+c)*(a^6-2*(b+c)*a^5-(b^2-8*b*c+c^2)*a^4+4*(b^2-c^2)*(b-c)*a^3-(b^4+c^4+2*b*c*(4*b^2-5*b*c+4*c^2))*a^2-2*(b+c)*(b^4+c^4-4*b*c*(b^2-b*c+c^2))*a+(b^4+c^4+2*b*c*(b^2-b*c+c^2))*(b-c)^2) : :

X(14878) lies on these lines: {3,1323}, {36,269}, {347,2071}, {905,6180}, {934,8069}, {1358,1470}

X(14878) = circumcircle-inverse-of-X(1323)

X(14879) = SIMPEDAL POINT OF ABC IN MIXTILINEAR TRIANGLE

Barycentrics    a^2*(a^10-(9*b^2-4*b*c+9*c^2)*a^8+8*(b+c)*(b^2+c^2)*a^7+2*(9*b^2-2*b*c+9*c^2)*(b-c)^2*a^6-4*(b+c)*(6*b^4+6*c^4-b*c*(8*b^2-11*b*c+8*c^2))*a^5-2*(5*b^6+5*c^6-(32*b^4+32*c^4-b*c*(33*b^2-28*b*c+33*c^2))*b*c)*a^4+8*(b+c)*(3*b^6+3*c^6-4*(2*b^4+2*c^4-b*c*(2*b^2-b*c+2*c^2))*b*c)*a^3-3*(b^6+c^6+(10*b^4+10*c^4+b*c*(7*b^2+12*b*c+7*c^2))*b*c)*(b-c)^2*a^2-4*(b^2-c^2)*(b-c)*(2*b^6+2*c^6-(4*b^4+4*c^4+b*c*(3*b^2+2*b*c+3*c^2))*b*c)*a+(b^2-c^2)^2*(b-c)^2*(3*b^4+3*c^4+2*b*c*(b^2+b*c+c^2))) : :

X(14879) lies on these lines: {}

X(14880) = SIMPEDAL POINT OF ABC IN 1st NEUBERG TRIANGLE

Barycentrics    a^8-2*b^2*c^2*a^4-(b^6+c^6)*a^2-(b^2-c^2)^2*b^2*c^2 : :
X(14880) = 3*X(3)-X(1975) = 3*X(381)-5*X(7851) = 3*X(549)-2*X(7789)

Let A'B'C' be the 1st Neuberg triangle and A"B"C" the 2nd Neuberg triangle. Let A* be the isogonal conjugate, wrt A'B'C', of A", and define B* and C* cyclically. The lines A'A*, B'B*, C'C* concur in X(14880). (Randy Hutson, November 2, 2017)

X(14880) lies on these lines: {2,10131}, {3,76}, {4,3398}, {5,182}, {6,14881}, {20,2080}, {30,32}, {83,381}, {140,7822}, {265,12192}, {355,12197}, {376,7793}, {382,11842}, {385,7470}, {511,7805}, {542,626}, {546,10358}, {548,8722}, {549,7789}, {550,5171}, {698,3098}, {1478,10799}, {1479,12835}, {1656,7943}, {1691,3767}, {2456,5207}, {2548,5038}, {2896,11177}, {3091,10359}, {3095,5999}, {3146,10788}, {3406,11606}, {3564,13355}, {3583,10798}, {3585,10797}, {3830,12150}, {3934,4048}, {4027,5025}, {5182,11318}, {5319,12212}, {5787,12196}, {6054,7899}, {6055,7749}, {6179,9301}, {6221,13885}, {6284,10801}, {6308,8667}, {6321,12176}, {6398,13938}, {6643,14152}, {6655,9862}, {6656,9996}, {6704,10168}, {7354,10802}, {7728,13193}, {7790,9873}, {7817,11645}, {7833,14830}, {7841,10350}, {7856,9993}, {7887,10352}, {7891,8724}, {7901,10334}, {7913,10356}, {7914,11178}, {7918,10347}, {7923,10345}, {7933,10333}, {10733,12201}, {10738,12199}, {10742,13194}, {10749,12207}, {11272,13860}, {12193,12293}, {12194,12699}, {12202,14216}, {12918,13195}, {13049,13050}

X(14880) = midpoint of X(5989) and X(12188)
X(14880) = reflection of X(4048) in X(5092)
X(14880) = 1st-Neuberg-isogonal conjugate of X(3)
X(14880) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 98, 10104), (3, 12188, 76), (4, 3398, 10796), (382, 11842, 12110), (385, 7470, 9821), (1676, 1677, 3818), (3818, 7834, 5), (4027, 5025, 10349), (7884, 14458, 381)

X(14881) = SIMPEDAL POINT OF ABC IN 2nd NEUBERG TRIANGLE

Barycentrics    (b^2+c^2)*a^6+(b^4+4*b^2*c^2+c^4)*a^4-(b^2+c^2)*(2*b^4-3*b^2*c^2+2*c^4)*a^2-(b^2-c^2)^2*b^2*c^2 : :
Barycentrics    sin 2B + sin B cos(B + 2ω) + sin 2C + sin C cos(C + 2ω) : :
X(14881) = X(3)-3*X(262) = 3*X(3)-5*X(7786) = 3*X(4)+X(194) = 3*X(5)-2*X(3934) = X(20)-3*X(11171) = X(76)-3*X(381) = X(194)-3*X(3095) = 9*X(262)-5*X(7786) = 3*X(262)-2*X(11272) = 5*X(7786)-6*X(11272)

Let A'B'C' be the 1st Neuberg triangle and A"B"C" the 2nd Neuberg triangle. Let A* be the isogonal conjugate, wrt A"B"C", of A', and define B* and C* cyclically. The lines A"A*, B"B*, C"C* concur in X(14881). (Randy Hutson, November 2, 2017)

X(14881) lies on these lines: {2,9821}, {3,83}, {4,147}, {5,141}, {6,14880}, {13,3104}, {14,3105}, {20,11171}, {30,39}, {32,2023}, {76,381}, {140,5188}, {315,9996}, {382,11257}, {538,3845}, {542,7838}, {546,6248}, {549,6683}, {550,13334}, {732,3818}, {736,7843}, {1078,9301}, {1478,12836}, {1479,12837}, {1656,7944}, {2548,3094}, {3070,3102}, {3071,3103}, {3090,6194}, {3091,7697}, {3098,7808}, {3146,7709}, {3329,7470}, {3398,5999}, {3583,13077}, {3767,6034}, {3830,7757}, {3843,13108}, {3860,14711}, {5052,5305}, {5066,9466}, {5149,9737}, {5476,7834}, {5969,7775}, {5976,7752}, {6287,9866}, {6309,9766}, {7760,12188}, {7812,9873}, {7818,10356}, {7896,11178}, {7900,9983}, {7941,9865}, {9818,9917}, {9955,12263}, {10063,10895}, {10079,10896}, {10104,13860}, {12699,12782}

X(14881) = midpoint of X(i) and X(j) for these {i,j}: {4, 3095}, {382, 11257}, {1916, 6033}, {3830, 7757}, {12699, 12782}
X(14881) = reflection of X(i) in X(j) for these (i,j): (3, 11272), (550, 13334), (3098, 10007), (5188, 140), (6248, 546), (9466, 5066), (12042, 2023), (12263, 9955)
X(14881) = complement of X(9821)
X(14881) = 2nd-Neuberg-isogonal conjugate of X(3)
X(14881) = intersection of Hatzipolakis axes of pedal triangles of PU(1)
X(14881) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 262, 11272), (3, 13111, 12110), (4, 7785, 6033), (3091, 12251, 7697), (3329, 7470, 12054), (7809, 14492, 381)

X(14882) = SIMPEDAL POINT OF 1st SCHIFFLER TRIANGLE IN ABC

Barycentrics    a^2*(a+b-c)*(a-b+c)*(a^3-(b+c)*a^2-(b^2+b*c+c^2)*a+(b+c)^3) : :

X(14882) lies on these lines: {1,3}, {11,6952}, {12,100}, {42,1399}, {43,7299}, {80,13743}, {108,5951}, {109,2594}, {197,9658}, {198,5341}, {497,6972}, {528,10957}, {603,2177}, {692,2477}, {759,14584}, {943,4995}, {1030,2171}, {1376,2476}, {1411,4642}, {1415,1500}, {1621,5433}, {1836,6796}, {2829,10955}, {3085,6951}, {3841,4413}, {3871,6224}, {4214,11383}, {4294,6903}, {4421,11237}, {5252,8715}, {5284,7294}, {5432,6853}, {5901,10090}, {6284,6840}, {6830,10896}, {6906,10950}, {6914,10573}, {6943,9670}, {7354,11491}, {8070,10738}, {10266,12519}, {10895,11499}, {10958,12764}, {11500,12943}, {12433,12832}, {12947,13146}

X(14882) = isogonal conjugate of X(10266)
X(14882) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 46, 5885), (1, 57, 13751), (1, 10284, 2098), (1, 11849, 55), (35, 65, 5172), (35, 484, 3), (55, 5204, 10267), (65, 5172, 56), (1388, 1470, 56), (1466, 11510, 56), (8071, 10679, 2098), (11248, 11507, 55), (11509, 11510, 1466)

X(14883) = SIMPEDAL POINT OF ABC IN 1st SCHIFFLER TRIANGLE

Barycentrics    a*(a^8-2*(2*b^2-b*c+2*c^2)*a^6+(6*b^2+b*c+6*c^2)*(b^2-b*c+c^2)*a^4+4*b^2*c^2*(b+c)*a^3-2*(2*b^4+2*c^4+b*c*(2*b^2+3*b*c+2*c^2))*(b-c)^2*a^2-(b^2-c^2)*(b-c)*b^2*c^2*a+(b^2-c^2)^2*(b-c)*(b^3-c^3))*(-a+b+c) : :

X(14883) lies on these lines: {11,13080}, {21,60}, {952,3065}, {4995,7161}, {6284,7701}

X(14884) = SIMPEDAL POINT OF ABC IN STEINER TRIANGLE

Barycentrics    (a^2-b^2)*(a^2-c^2)*(a^12-2*(b^2+c^2)*a^10+3*(b^4+c^4)*a^8-2*(b^2+c^2)*(2*b^4-3*b^2*c^2+2*c^4)*a^6+(3*b^8+3*c^8-(2*b^4+b^2*c^2+2*c^4)*b^2*c^2)*a^4-2*(b^6+c^6)*(b^2-c^2)^2*a^2+(b^4+c^4)*(b^2-c^2)^4) : :

X(14884) lies on these lines: {2,6328}, {523,14366}

X(14884) = anticomplement of X(6328)

X(14885) = SIMPEDAL POINT OF SYMMEDIAL TRIANGLE IN ABC

Barycentrics    a^2*(a^4-(b^2+c^2)*a^2+b^2*c^2-(b^2+c^2)^2)*(a^2+c^2)*(a^2+b^2) : :

Let A'B'C' be the symmedial triangle. Let Oa be the circumcenter of AB'C', and define Ob and Oc cyclically. Triangle OaObOc is similar to ABC, with similitude center X(14885). (Randy Hutson, November 2, 2017)

X(14885) lies on the cubic K629 and these lines: {6,14370}, {23,251}, {32,14247}, {39,827}, {83,115}, {384,4577}, {695,1176}, {733,3398}, {5299,7122}

X(14885) = isogonal conjugate of X(33665)
X(14885) = X(6)-Ceva conjugate of X(251)
X(14885) = Miquel associate of X(6)
X(14885) = {X(39), X(827)}-harmonic conjugate of X(9481)

X(14886) = SIMPEDAL POINT OF ABC IN SYMMEDIAL TRIANGLE

Barycentrics    a^2*(a^8-(b^2+c^2)*a^6-2*(b^2+c^2)^2*a^4-(b^4+b^2*c^2+c^4)*(b^2+c^2)*a^2+(b^4-b^2*c^2-c^4)*(b^4+b^2*c^2-c^4)) : :

X(14886) lies on the line {6,6655}

X(14887) = SIMPEDAL POINT OF YFF CONTACT TRIANGLE IN ABC

Barycentrics    a^2*(a^4-(b+c)*a^3+b*c*a^2+(b^2-c^2)*(b-c)*a-(b^3-c^3)*(b-c))*(-a+c)^2*(a-b)^2 : :

X(14887) lies on the cubic K165 and these lines: {1,59}, {4,5377}, {35,1110}, {72,765}, {78,6065}, {514,14888}, {1252,3730}, {1265,4076}, {1385,4564}, {1618,6161}, {9268,14260}, {13589,14513}

X(14887) = antigonal conjugate of X(150)

X(14888) = SIMPEDAL POINT OF ABC IN YFF CONTACT TRIANGLE

Barycentrics    (a^8-2*(b+c)*a^7+2*(b^2+b*c+c^2)*a^6-2*b*c*(b+c)*a^5-(3*b^4+3*c^4-b*c*(4*b^2-b*c+4*c^2))*a^4+3*(b^4-c^4)*(b-c)*a^3-(b^4+c^4+(b^2+b*c+c^2)*b*c)*(b-c)^2*a^2-(b^4-c^4)*(b-c)^3*a+(b^2-c^2)*(b-c)^3*(b^3+c^3))*(-a+c)*(a-b) : :

X(14888) lies on the line {514,14887}

X(14889) = SIMPEDAL POINT OF ABC IN YIU TRIANGLE

Barycentrics    a^2*(-a^2+b^2+c^2)*(a^12-4*(b^2+c^2)*a^10+(6*b^4+7*b^2*c^2+6*c^4)*a^8-(b^2+c^2)*(2*b^2-3*b*c+2*c^2)*(2*b^2+3*b*c+2*c^2)*a^6+(b^8+c^8)*a^4+(b^4-c^4)*(b^2-c^2)*b^2*c^2*a^2-(b^2-c^2)^4*b^2*c^2) : :

X(14889) lies on these lines: {3,49}, {110,8154}, {1141,3153}, {1154,5961}, {5962,7577}

X(14889) = circumcircle-inverse-of-X(49)
X(14889) = Yiu circle-inverse-of-X(110)

X(14890) = MIDPOINT OF X(140) AND X(5054)

Barycentrics    55*S^2-21*SB*SC : :
X(14890) = 19*X(2)-7*X(5) = X(2)-7*X(140) = 13*X(2)-7*X(547) = 5*X(2)+X(548) = 5*X(2)+7*X(549) = 9*X(2)+7*X(3524) = 8*X(2)+7*X(3530) = 23*X(2)-7*X(3545) = 13*X(2)-X(3627) = 10*X(2)-7*X(3628) = 4*X(2)-X(3850) = X(2)+7*X(5054) = 15*X(2)-7*X(5055) = 25*X(2)-7*X(5066) = 16*X(2)-7*X(10109) = 4*X(2)-7*X(10124) = 3*X(2)-7*X(11539) = 5*X(2)-14*X(11540) = 22*X(2)-7*X(11737) = 2*X(2)+7*X(11812)

As a point on the Euler line, X(14890) has Shinagawa coefficients (55, -21)

See Antreas Hatzipolakis and César Lozada, Hyacinthos 26695.

X(14890) lies on this line: {2, 3}

X(14890) = midpoint of X(i) and X(j) for these {i,j}: {140, 5054}, {5066, 10304}
X(14890) = reflection of X(i) in X(j) for these (i,j): (3861, 3545), (11812, 5054), (14269, 12811)
X(14890) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 3627, 547), (4, 549, 12100), (472, 7924, 4201), (1346, 7462, 3144), (3363, 7460, 3145), (3843, 5059, 3627), (3850, 10124, 2), (3850, 12108, 3530), (5071, 6948, 8703), (7450, 7458, 6894), (7563, 8370, 13729), (11305, 13729, 7395), (12812, 14869, 12108)

X(14891) = EULER LINE INTERCEPT OF X(3633)X(3654)

Barycentrics    23*S^2-21*SB*SC : :
X(14891) = X(2)+7*X(3) = 23*X(2)-7*X(4) = 11*X(2)-7*X(5) = 5*X(2)-7*X(140) = 9*X(2)+7*X(376) = 15*X(2)-7*X(381) = 17*X(2)-7*X(546) = 9*X(2)-7*X(547) = 3*X(2)-7*X(549) = 13*X(2)+7*X(550) = 7*X(2)+X(1657) = 2*X(2)-7*X(3530) = 5*X(2)-X(3627) = 8*X(2)-7*X(3628) = 13*X(2)-5*X(3843) = 19*X(2)-7*X(3845) = 31*X(2)-14*X(3856) = 16*X(2)-7*X(3860) = X(3633)+7*X(3654)

As a point on the Euler line, X(14891) has Shinagawa coefficients (23, -21)

See Antreas Hatzipolakis and César Lozada, Hyacinthos 26695.

X(14891) lies on these lines: {2, 3}, {3633, 3654}, {4114, 5719}, {5355, 8589}, {6053, 13392}, {11694, 12041}

X(14891) = midpoint of X(i) and X(j) for these {i,j}: {2, 548}, {3, 12100}, {20, 12101}, {140, 8703}, {376, 547}, {546, 3534}, {550, 5066}, {3845, 12103}, {11694, 12041}
X(14891) = reflection of X(i) in X(j) for these (i,j): (2, 12108), (5, 11540), (3530, 12100), (3628, 11812), (3830, 3856), (3850, 2), (3860, 3628), (3861, 10109), (10109, 140), (10124, 549), (11737, 10124), (11812, 3530), (12101, 12811), (12102, 5066)
X(14891) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (140, 546, 5070), (140, 3861, 3628), (381, 549, 140), (381, 3524, 549), (381, 5070, 5071), (381, 10109, 11737), (3090, 3543, 381), (3530, 3850, 12108), (3627, 3850, 3861), (3843, 5054, 2), (3843, 5072, 3854), (3850, 12102, 3843), (3861, 11737, 381)

X(14892) = EULER LINE INTERCEPT OF X(3630)X(11178)

Barycentrics    17*S^2+21*SB*SC : :
X(14892) = 19*X(2)-7*X(3) = X(2)-7*X(5) = 10*X(2)-7*X(140) = 5*X(2)+7*X(381) = 8*X(2)+7*X(546) = 4*X(2)-7*X(547) = 4*X(2)-X(548) = 13*X(2)-7*X(549) = 13*X(2)-X(1657) = 15*X(2)-7*X(3524) = 29*X(2)-14*X(3530) = X(2)+7*X(3545) = 5*X(2)+X(3627) = 11*X(2)-14*X(3628) = 9*X(2)+7*X(3839) = 7*X(2)+5*X(3843) = 11*X(2)+7*X(3845) = X(2)+2*X(3850) = 11*X(2)-7*X(5054) = 3*X(2)-7*X(5055)

As a point on the Euler line, X(14892) has Shinagawa coefficients (17, 21)

See Antreas Hatzipolakis and César Lozada, Hyacinthos 26695.

X(14892) lies on these lines: {2, 3}, {3630, 11178}, {3656, 4668}, {4691, 9955}, {7687, 11694}, {10113, 11693}

X(14892) = midpoint of X(i) and X(j) for these {i,j}: {5, 3545}, {549, 14269}, {3839, 11539}, {3845, 5054}, {10113, 11693}
X(14892) = reflection of X(i) in X(j) for these (i,j): (3545, 11737), (3853, 14269), (5054, 3628), (5066, 3545), (10304, 11812), (12103, 10304), (14269, 3860)
X(14892) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 4, 14093), (4, 8226, 7413), (5, 3845, 5071), (140, 3627, 548), (381, 12101, 546), (382, 3851, 13587), (546, 547, 12100), (547, 12101, 140), (1567, 6862, 4234), (3090, 3861, 140), (3627, 5072, 12811), (3839, 5055, 11539), (3845, 5071, 3628)

X(14893) =  EULER LINE INTERCEPT OF X(517)X(4532)

Barycentrics    S^2+21*SB*SC : :
11*X(2)-7*X(3) = X(2)+7*X(4) = 5*X(2)-7*X(5) = 23*X(2)-7*X(20) = 8*X(2)-7*X(140) = 15*X(2)-7*X(376) = 3*X(2)-7*X(381) = 13*X(2)+7*X(382) = 2*X(2)-7*X(546) = 6*X(2)-7*X(547) = 9*X(2)-7*X(549) = 17*X(2)-7*X(550) = 5*X(2)-X(1657) = 19*X(2)-14*X(3530) = 19*X(2)-7*X(3534) = 9*X(2)+7*X(3543) = 13*X(2)-14*X(3628) = 5*X(2)+7*X(3830) = X(3630)-7*X(3818) = 5*X(4668)+7*X(12699)

As a point on the Euler line, X(14893) has Shinagawa coefficients (1, 21)

See Antreas Hatzipolakis and César Lozada, Hyacinthos 26695.

X(14893) lies on these lines:
{2, 3}, {265, 14487}, {397, 12816}, {398, 12817}, {517, 4532}, {541, 11801}, {1587, 6499}, {1588, 6498}, {3163, 6748}, {3630, 3818}, {4668, 12699}, {5305, 14537}, {5309, 14075}, {5480, 13687}, {5663, 13451}, {10095, 13474}, {13364, 13570}

X(14893) = midpoint of X(i) and X(j) for these {i,j}: {2, 3627}, {4, 3845}, {5, 3830}, {382, 8703}, {546, 12101}, {549, 3543}, {3853, 5066}, {3860, 12102}
X(14893) = reflection of X(i) in X(j) for these (i,j): (2, 3850), (3, 10109), (140, 5066), (376, 10124), (546, 3845), (547, 381), (548, 2), (549, 11737), (550, 11812), (3534, 3530), (3830, 12102), (3845, 3861), (3853, 12101), (5066, 546), (8703, 3628), (10109, 3856), (12100, 5), (12101, 4), (12103, 12100), (13364, 13570)
X(14893) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (5, 3627, 1657), (381, 3543, 549), (382, 3545, 8703), (382, 3858, 3628), (546, 3627, 12812), (1658, 1885, 3529), (3627, 3843, 3850), (3627, 3850, 548), (3830, 3839, 5), (3830, 3845, 3860), (3830, 3860, 12100), (3855, 5073, 632), (5073, 6885, 12108)

X(14894) =  EULER LINE INTERCEPT OF X(389)X(523)

Barycentrics    2*S^4-(-2*SW^2+(12*R^2+3*SA)* SW-3*SA^2)*S^2+(-3*SW^2+34*R^ 2*SW-72*R^4)*(SB+SC)*SA : :

See Antreas Hatzipolakis and César Lozada, Hyacinthos 26696.

X(14894) lies on these lines: {2, 3}, {389, 523}, {2452, 11432}, {2453, 9786}, {3258, 3574}, {10095, 12052}

X(14894) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (381, 10124, 7483), (6893, 11548, 3560), (7394, 8226, 11818), (7450, 13163, 4240)

X(14895) =  EULER LINE INTERCEPT OF X(523)X(12241)

Barycentrics    2*S^4-(6*SW^2+(-56*R^2+3*SA)* SW+144*R^4-3*SA^2)*S^2+(144*R^ 4-68*R^2*SW+9*SW^2)*(SB+SC)*SA : :

See Antreas Hatzipolakis and César Lozada, Hyacinthos 26696.

X(14895) lies on these lines: {2, 3}, {523, 12241}, {5462, 12052}

X(14895) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1316, 7491, 452), (6936, 11292, 6940)

X(14896) =  (name pending)

Barycentrics    2*S^4-(2*SW^2+(-22*R^2+3*SA)* SW+72*R^4-3*SA^2)*S^2+(36*R^4- 17*R^2*SW+3*SW^2)*(SB+SC)*SA : :

See Antreas Hatzipolakis and César Lozada, Hyacinthos 26696.

X(14896) lies on this line: {2, 3}

X(14896) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1314, 6840, 1981), (4199, 14093, 426), (4208, 7412, 1583), (4224, 11288, 3523), (6846, 7532, 7562), (6921, 11335, 471), (7401, 11007, 13747), (8368, 12057, 4204), (10128, 10565, 2676)

X(14897) =  (name pending)

Barycentrics    3*a^4*b^4 + 26*a^4*b^2*c^2 - 2*a^2*b^4*c^2 + 3*a^4*c^4 - 2*a^2*b^2*c^4 - b^4*c^4 : :

X(14897) lies on this line: {4664,17448}

X(14898) =  X(2)X(6)∩X(111)X(351)

Barycentrics    a^2*(a^6*b^2 + 2*a^4*b^4 + a^2*b^6 + a^6*c^2 - 10*a^4*b^2*c^2 + 2*a^2*b^4*c^2 - 5*b^6*c^2 + 2*a^4*c^4 + 2*a^2*b^2*c^4 + 8*b^4*c^4 + a^2*c^6 - 5*b^2*c^6) : :

X(14898) lies on the cubic K138 and these lines: {2, 6}, {3, 14684}, {98, 729}, {111, 351}, {187, 11634}, {842, 14659}, {843, 9136}, {3291, 5968}, {5106, 9149}, {6041, 7418}

X(14898) = crosspoint of X(111) and X(5970)
X(14898) = crossdifference of every pair of points on line {512, 2482}
X(14898) = crosssum of X(524) and X(5969)
X(14898) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6, 3231, 2421), (187, 14609, 11634), (2086, 3231, 6)

X(14899) =  X(2)X(1341)∩X(6)X(1345)

Barycentrics    a*(-(a*Sqrt(a^4 - a^2*b^2 + b^4 - a^2*c^2 - b^2*c^2 + c^4)*(a^6*b^2 - a^4*b^4 - a^2*b^6 + b^8 + a^6*c^2 - a^4*b^2*c^2 + 3*a^2*b^4*c^2 + b^6*c^2 - a^4*c^4 + 3*a^2*b^2*c^4 - 4*b^4*c^4 - a^2*c^6 + b^2*c^6 + c^8)) - a*(a^6*b^4 - a^4*b^6 - a^2*b^8 + b^10 + 2*a^6*b^2*c^2 - 3*a^4*b^4*c^2 + 4*a^2*b^6*c^2 + b^8*c^2 + a^6*c^4 - 3*a^4*b^2*c^4 - 2*a^2*b^4*c^4 - 2*b^6*c^4 - a^4*c^6 + 4*a^2*b^2*c^6 - 2*b^4*c^6 - a^2*c^8 + b^2*c^8 + c^10) + Sqrt(a^6 - a^4*b^2 - a^2*b^4 + b^6 - a^4*c^2 + 3*a^2*b^2*c^2 - b^4*c^2 - a^2*c^4 - b^2*c^4 + c^6)*(-(b*c*(a^2 + b^2 - 2*b*c + c^2)*(a^2 + b^2 + 2*b*c + c^2)*Sqrt(a^4 - a^2*b^2 + b^4 - a^2*c^2 - b^2*c^2 + c^4)) - b*c*(a^6 - a^4*b^2 + 3*a^2*b^4 + b^6 - a^4*c^2 - 2*a^2*b^2*c^2 - b^4*c^2 + 3*a^2*c^4 - b^2*c^4 + c^6))) : :

X(14899) lies on the cubics K138, K881, K886j, K887, and these lines: {2, 1341}, {6, 1345}, {111, 2469}, {1114, 1379}

X(14899) = crossdifference of every pair of points on line {2574, 5639}

X(14900) =  X(4)X(32)∩X(20)X(648)

Barycentrics    (a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*(4*a^10 - 10*a^8*b^2 + 9*a^6*b^4 - 3*a^4*b^6 - a^2*b^8 + b^10 - 10*a^8*c^2 + 10*a^6*b^2*c^2 - 3*a^4*b^4*c^2 + 4*a^2*b^6*c^2 - b^8*c^2 + 9*a^6*c^4 - 3*a^4*b^2*c^4 - 6*a^2*b^4*c^4 - 3*a^4*c^6 + 4*a^2*b^2*c^6 - a^2*c^8 - b^2*c^8 + c^10) : :
X(14900) = X(4) - 3 X(112), 2 X(4) - 3 X(132), 3 X(127) - 4 X(140), 3 X(1297) - 5 X(3522), 5 X(1656) - 6 X(6720), 5 X(631) - 3 X(10718), 5 X(4) - 3 X(10735), 5 X(132) - 2 X(10735), 5 X(112) - X(10735), 5 X(1656) - 3 X(10749), X(5059) + 3 X(12384), X(5073) - 3 X(12918), X(132) + 2 X(13200), X(4) + 3 X(13200), X(10735) + 5 X(13200), 7 X(3523) - 3 X(13219), X(1657) + 3 X(13310), 3 X(11722) - 2 X(13464), 2 X(550) - 3 X(14689)

X(14900) lies on the cubic K824 and these lines: {4, 32}, {20, 648}, {127, 140}, {185, 1205}, {550, 14689}, {631, 10718}, {1297, 3522}, {1656, 6720}, {1657, 13310}, {2799, 10992}, {2806, 10993}, {2848, 9409}, {3517, 11641}, {3523, 13219}, {3575, 8754}, {4232, 9157}, {5059, 12384}, {5073, 12918}, {8960, 13923}, {11722, 13464}

X(14900) = midpoint of X(112) and X(13200)
X(14900) = reflection of X(i) in X(j) for these {i,j}: {132, 112}, {10749, 6720}
X(14900) = polar-circle-inverse of X(14639)

X(14901) =  MIDPOINT OF X(12375) AND X(12376)

Barycentrics    a^2*(a^8 - 2*a^6*b^2 + 2*a^4*b^4 - 2*a^2*b^6 + b^8 - 2*a^6*c^2 + a^4*b^2*c^2 + a^2*b^4*c^2 + b^6*c^2 + 2*a^4*c^4 + a^2*b^2*c^4 - 4*b^4*c^4 - 2*a^2*c^6 + b^2*c^6 + c^8) : :
X(14901) = X(74) - 3 X(11653)

X(14901) lies on the cubic K290 and these lines: {6, 13}, {32, 5663}, {39, 110}, {74, 187}, {125, 7746}, {146, 7737}, {230, 10264}, {323, 538}, {526, 2510}, {574, 1511}, {577, 12358}, {690, 2422}, {895, 1570}, {1015, 10091}, {1500, 10088}, {1506, 14643}, {1555, 12112}, {1692, 11579}, {1914, 7727}, {1968, 12292}, {1971, 13289}, {1986, 10311}, {2241, 3024}, {2242, 3028}, {2549, 12383}, {2854, 5028}, {2948, 9620}, {3053, 10620}, {3094, 12584}, {3448, 3767}, {3815, 10272}, {5007, 14094}, {5034, 6593}, {5052, 9970}, {5206, 12041}, {5286, 14683}, {5609, 7772}, {7722, 10312}, {7723, 10316}, {7724, 10315}, {7728, 7747}, {7735, 12317}, {7739, 9143}, {7756, 12121}, {9604, 11597}, {9664, 12896}, {9826, 10314}, {10313, 12219}, {10706, 14537}

X(14901) = midpoint of X(12375) and X(12376)
X(14901) = crossdifference of every pair of points on line {526, 12824}
X(14901) = X(2510)-line conjugate of X(526)

X(14902) =  X(13)X(15)∩X(476)X(755)

Barycentrics    2*S*(a^8 + 4*a^6*b^2 - 4*a^4*b^4 - 2*a^2*b^6 + b^8 + 4*a^6*c^2 + a^4*b^2*c^2 + a^2*b^4*c^2 - 2*b^6*c^2 - 4*a^4*c^4 + a^2*b^2*c^4 + 2*b^4*c^4 - 2*a^2*c^6 - 2*b^2*c^6 + c^8) - Sqrt(3)*(a^10 - 3*a^8*b^2 + 4*a^4*b^6 - a^2*b^8 - b^10 - 3*a^8*c^2 - 3*a^6*b^2*c^2 + a^2*b^6*c^2 + 3*b^8*c^2 - 2*b^6*c^4 + 4*a^4*c^6 + a^2*b^2*c^6 - 2*b^4*c^6 - a^2*c^8 + 3*b^2*c^8 - c^10) : :

X(14902) lies on the cubic K290 and these lines: {13, 15}, {476, 755}, {538, 11078}

X(14903) =  X(14)X(16)∩X(476)X(755)

Barycentrics    2*S*(a^8 + 4*a^6*b^2 - 4*a^4*b^4 - 2*a^2*b^6 + b^8 + 4*a^6*c^2 + a^4*b^2*c^2 + a^2*b^4*c^2 - 2*b^6*c^2 - 4*a^4*c^4 + a^2*b^2*c^4 + 2*b^4*c^4 - 2*a^2*c^6 - 2*b^2*c^6 + c^8) + Sqrt(3)*(a^10 - 3*a^8*b^2 + 4*a^4*b^6 - a^2*b^8 - b^10 - 3*a^8*c^2 - 3*a^6*b^2*c^2 + a^2*b^6*c^2 + 3*b^8*c^2 - 2*b^6*c^4 + 4*a^4*c^6 + a^2*b^2*c^6 - 2*b^4*c^6 - a^2*c^8 + 3*b^2*c^8 - c^10) : :

X(14903) lies on the cubic K290 and these lines: {14, 16}, {476, 755}, {538, 11092}

X(14904) =  X(2)X(13)∩X(99)X(141)

Barycentrics    -2*a^4 + 3*a^2*b^2 + b^4 + 3*a^2*c^2 + c^4 + Sqrt(3)*2*S*(b^2 + c^2) : :

X(14904) lies on the cubic K290 and these lines: {2, 13}, {39, 6297}, {99, 141}, {115, 302}, {298, 538}, {299, 7810}, {395, 7827}, {473, 12142}, {542, 11299}, {599, 11129}, {623, 14041}, {635, 6655}, {636, 7824}, {3642, 7833}, {5617, 6248}, {5969, 11128}, {8594, 11161}, {11305, 13103}

X(14904) = circumcircle-of-inner-Napoleon-triangle-inverse of X(6582)
X(14904) = crossdifference of every pair of points on line {6137, 14428}
X(14904) = psi-transform of X(6299)
X(14904) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (13, 5463, 6582), (141,8356,14905), (6302, 6306, 16)

X(14905) =  X(2)X(14)∩X(99)X(141)

Barycentrics    2*a^4 - 3*a^2*b^2 - b^4 - 3*a^2*c^2 - c^4 + Sqrt(3)*2*S*(b^2 + c^2) : :

X(14905) lies on the cubic K290 and these lines: {2, 14}, {39, 6296}, {99, 141}, {115, 303}, {298, 7810}, {299, 538}, {396, 7827}, {472, 12141}, {542, 11300}, {599, 11128}, {624, 14041}, {635, 7824}, {636, 6655}, {3643, 7833}, {5613, 6248}, {5969, 11129}, {8595, 11161}, {11306, 13102}

X(14905) = crossdifference of every pair of points on line {6138, 14428}
X(14905) = circumcircle-of-inner-Napoleon-triangle-inverse of X(6295)
X(14905) = psi-transform of X(6298)
X(14905) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (14, 5464, 6295), (141,8356,14904), (6303, 6307, 15)

X(14906) =  X(30)X(599)∩X(574)X(1495)

Barycentrics    a^2/(-3*a^4 + 2*a^2*b^2 + b^4 + 2*a^2*c^2 + c^4) : :

X(14906) lies on the cubic K298 and these lines: {30, 599}, {574, 1495}, {2871, 11173}, {5052, 8541}

X(14906) = isogonal conjugate of X(14907)
X(14906) = X(75)-isoconjugate of X(6800)
X(14906) = barycentric quotient X(32)/X(6800)

X(14907) =  X(2)X(187)∩X(3)X(315)

Barycentrics    -3*a^4 + 2*a^2*b^2 + b^4 + 2*a^2*c^2 + c^4 : :

X(14907) lies on these lines: {2, 187}, {3, 315}, {4, 1078}, {5, 9754}, {6, 8356}, {20, 76}, {30, 183}, {32, 4045}, {69, 74}, {140, 7773}, {141, 1003}, {193, 7757}, {194, 14023}, {230, 7841}, {274, 4190}, {302, 11300}, {303, 11299}, {317, 378}, {350, 4302}, {384, 7800}, {385, 2549}, {439, 7922}, {489, 11825}, {490, 11824}, {491, 6200}, {492, 6396}, {548, 3933}, {550, 1975}, {574, 754}, {577, 4611}, {599, 8598}, {620, 7818}, {626, 5206}, {631, 7752}, {637, 9739}, {638, 9738}, {671, 11172}, {691, 2857}, {1007, 3524}, {1285, 3618}, {1352, 11676}, {1369, 14558}, {1370, 1799}, {1384, 7792}, {1909, 4299}, {2080, 9753}, {2482, 7908}, {2548, 7823}, {2794, 8722}, {2896, 3552}, {3053, 6656}, {3096, 14001}, {3314, 13586}, {3522, 3926}, {3523, 7769}, {3528, 6337}, {3619, 14039}, {3734, 6781}, {3760, 4324}, {3761, 4316}, {3763, 6661}, {3767, 6655}, {3788, 7873}, {3793, 8354}, {3934, 14035}, {4234, 5224}, {5013, 7762}, {5023, 7784}, {5210, 7778}, {5286, 6179}, {5304, 7827}, {5319, 7864}, {5939, 14830}, {6390, 7788}, {6680, 7935}, {7495, 11187}, {7610, 8352}, {7615, 8597}, {7618, 7840}, {7735, 7790}, {7736, 7812}, {7738, 7760}, {7739, 7766}, {7745, 11285}, {7746, 7842}, {7747, 7815}, {7748, 7780}, {7749, 7825}, {7751, 7756}, {7755, 7872}, {7758, 7783}, {7779, 9939}, {7781, 7826}, {7789, 7879}, {7799, 7850}, {7801, 7848}, {7806, 7924}, {7816, 7854}, {7820, 7865}, {7822, 14037}, {7828, 7910}, {7832, 7936}, {7835, 7883}, {7836, 7929}, {7845, 8589}, {7851, 8357}, {7857, 7911}, {7863, 7896}, {7868, 8369}, {7885, 7907}, {7891, 7939}, {7892, 7928}, {7944, 14069}, {8353, 8667}, {8359, 11174}, {8721, 9863}, {9740, 11054}, {10298, 13219}, {10788, 14561}, {11168, 11317}

X(14907) = anticomplement X(5475)
X(14907) = reflection of X(i) in X(j) for these {i,j}: {7774, 574}, {9744, 3}, {11185, 183}, {11317, 11168}
X(14907) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 14712, 7737), (3, 315, 7763), (3, 7750, 315), (20, 3785, 76), (32, 7791, 7803), (32, 7830, 7791), (69, 376, 99), (99, 7811, 69), (187, 7761, 2), (316, 7771, 2), (384, 7904, 7800), (385, 7833, 2549), (550, 7767, 1975), (1078, 7802, 4), (1285, 3618, 12150), (1384, 11287, 7792), (2896, 3552, 7795), (3522, 3926, 7782), (3972, 7831, 2), (5023, 7784, 7807), (6179, 7847, 5286), (6655, 7793, 3767), (6781, 7810, 3734), (7746, 7842, 14063), (7768, 7782, 3926), (7771, 11057, 316), (7783, 7893, 7758), (7818, 8588, 620), (7823, 7824, 2548), (7857, 7911, 14064)
X(14907) = isotomic conjugate X(14906)
X(14907) = barycentric product X(76)*X(6800)
X(14907) = barycentric quotient X(6800)/X(6)

X(14908) =  X(25)X(111)∩X(30)X(98)

Barycentrics    a^4*(a^2 + b^2 - 2*c^2)*(a^2 - b^2 - c^2)*(a^2 - 2*b^2 + c^2) : :

X(14908) lies on the cubic K478 and these lines: {3, 895}, {25, 111}, {30, 98}, {32, 1084}, {187, 2393}, {228, 906}, {248, 878}, {351, 2444}, {468, 10422}, {682, 10547}, {923, 1402}, {1368, 1799}, {1976, 2420}, {2353, 3053}, {2482, 9145}, {2854, 2936}, {3425, 5968}, {8263, 8369}, {10558, 14586}, {12106, 14246}

X(14908) = isogonal conjugate of the isotomic conjugate of X(895)
X(14908) = X(1726)-zayin conjugate of X(896)
X(14908) = X(i)-Ceva conjugate of X(j) for these (i,j): {691, 10097}, {10422, 6}
X(14908) = X(10317)-cross conjugate of X(10547)
X(14908) = X(i)-isoconjugate of X(j) for these (i,j): {4, 14210}, {19, 3266}, {75, 468}, {92, 524}, {158, 6390}, {187, 1969}, {264, 896}, {273, 3712}, {286, 4062}, {318, 7181}, {690, 811}, {799, 14273}, {823, 14417}, {1577, 4235}, {2642, 6331}, {4750, 6335}
X(14908) = crosspoint of X(111) and X(895)
X(14908) = trilinear pole of line {184, 3049}
X(14908) = crosssum of X(i) and X(j) for these (i,j): {468, 524}, {2393, 3291}
X(14908) = crossdifference of every pair of points on line X(126)X(1560)
X(14908) = barycentric product of circumcircle intercepts of line X(3)X(647)
X(14908) = barycentric product X(i)*X(j) for these {i,j}: {3, 111}, {6, 895}, {48, 897}, {63, 923}, {110, 10097}, {184, 671}, {219, 7316}, {222, 5547}, {248, 5968}, {394, 8753}, {647, 691}, {892, 3049}, {3284, 9139}, {3289, 9154}, {3292, 10630}, {4558, 9178}, {6091, 8770}, {10317, 10415}
X(14908) = barycentric quotient X(i)/X(j) for these {i,j}: {3, 3266}, {32, 468}, {48, 14210}, {111, 264}, {184, 524}, {577, 6390}, {669, 14273}, {691, 6331}, {895, 76}, {897, 1969}, {923, 92}, {1576, 4235}, {2200, 4062}, {3049, 690}, {5547, 7017}, {7316, 331}, {8753, 2052}, {9178, 14618}, {9247, 896}, {10097, 850}, {10317, 7664}, {14567, 5095}, {14575, 187}, {14585, 3292}, {14600, 5967}

X(14909) =  X(23)X(895)∩X(935)X(14833)

Barycentrics    a^2*(a^2 + b^2 - 2*c^2)*(a^2 - b^2 - c^2)*(a^2 - 2*b^2 + c^2)*(a^12 + a^10*b^2 - 3*a^8*b^4 - 2*a^6*b^6 + 3*a^4*b^8 + a^2*b^10 - b^12 + a^10*c^2 - a^8*b^2*c^2 + 4*a^6*b^4*c^2 - 5*a^2*b^8*c^2 + b^10*c^2 - 3*a^8*c^4 + 4*a^6*b^2*c^4 - 6*a^4*b^4*c^4 + 4*a^2*b^6*c^4 + b^8*c^4 - 2*a^6*c^6 + 4*a^2*b^4*c^6 - 2*b^6*c^6 + 3*a^4*c^8 - 5*a^2*b^2*c^8 + b^4*c^8 + a^2*c^10 + b^2*c^10 - c^12) : :

X(14909) lies on the cubic K335 and these lines: {23, 895}, {935, 14833}

X(14909) = X(22)-Ceva conjugate of X(6091)

X(14910) =  ISOGONAL CONJUGATE OF X(3580)

Trilinears    (sin A)/(1 + cos 2B + cos 2C) : :
Barycentrics    a^2*(a^6 - a^4*b^2 - a^2*b^4 + b^6 - 2*a^4*c^2 + 2*a^2*b^2*c^2 - 2*b^4*c^2 + a^2*c^4 + b^2*c^4)*(a^6 - 2*a^4*b^2 + a^2*b^4 - a^4*c^2 + 2*a^2*b^2*c^2 + b^4*c^2 - a^2*c^4 - 2*b^2*c^4 + c^6) : :

X(14910) lies on the cubic K505, the circumconic {A,B,C,X(2),X(6)} and these lines: {2, 2986}, {6, 1511}, {25, 1576}, {30, 50}, {32, 3163}, {37, 906}, {111, 10313}, {112, 393}, {115, 577}, {186, 3003}, {230, 8791}, {248, 2395}, {526, 686}, {687, 2966}, {1415, 1880}, {1976, 2393}, {2963, 9722}, {6103, 13854}, {8882, 14586}, {10311, 10418}

Let A'B'C' be the circumcevian triangle of X(30). Let A" be the barycentric product B'*C', and define B" and C" cyclically. A", B", C" are collinear on line X(1637)X(3284). The lines AA", BB", CC" concur in X(14910). (Randy Hutson, August 19, 2019)

X(14910) = isogonal conjugate of X(3580)
X(14910) = X(i)-zayin conjugate of X(j) for these (i,j): {1, 3580}, {1726, 2315}
X(14910) = X(2986)-Ceva conjugate of X(5504)
X(14910) = X(i)-cross conjugate of X(j) for these (i,j): {1637, 112}, {3284, 6}, {14270, 110}
X(14910) = X(i)-isoconjugate of X(j) for these (i,j): {1, 3580}, {2, 1725}, {63, 403}, {75, 3003}, {92, 13754}, {113, 2349}, {162, 6334}, {264, 2315}, {686, 811}, {14206, 14264}
X(14910) = X(523)-vertex conjugate of X(1177)
X(14910) = cevapoint of X(i) and X(j) for these (i,j): {6, 50}, {32, 1495}, {647, 2088}, {3124, 9409}
X(14910) = crosspoint of X(1300) and X(2986)
X(14910) = trilinear pole of line {184, 512}
X(14910) = crossdifference of every pair of points on line {113, 131}
X(14910) = crosssum of X(i) and X(j) for these (i,j): {6, 2931}, {2088, 9033}, {3003, 13754}
X(14910) = perspector of ABC and unary cofactor triangle of Ehrmann side-triangle
X(14910) = barycentric product of circumcircle intercepts of line X(3)X(523)
X(14910) = barycentric product X(i)*X(j) for these {i,j}: {3, 1300}, {4, 5504}, {6, 2986}, {30, 10419}, {186, 12028}, {523, 10420}, {647, 687}
X(14910) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 3580}, {25, 403}, {31, 1725}, {32, 3003}, {184, 13754}, {647, 6334}, {687, 6331}, {1300, 264}, {1495, 113}, {2393, 12827}, {2986, 76}, {3049, 686}, {5504, 69}, {9247, 2315}, {10419, 1494}, {10420, 99}, {12028, 328}

X(14911) =  X(30)X(2986)∩X(186)X(476)

Barycentrics    (a^6 - a^4*b^2 - a^2*b^4 + b^6 - 2*a^4*c^2 + 2*a^2*b^2*c^2 - 2*b^4*c^2 + a^2*c^4 + b^2*c^4)*(a^6 - 2*a^4*b^2 + a^2*b^4 - a^4*c^2 + 2*a^2*b^2*c^2 + b^4*c^2 - a^2*c^4 - 2*b^2*c^4 + c^6)*(a^10 + a^8*b^2 - 8*a^6*b^4 + 8*a^4*b^6 - a^2*b^8 - b^10 + a^8*c^2 + 9*a^6*b^2*c^2 - 6*a^4*b^4*c^2 - 7*a^2*b^6*c^2 + 3*b^8*c^2 - 8*a^6*c^4 - 6*a^4*b^2*c^4 + 16*a^2*b^4*c^4 - 2*b^6*c^4 + 8*a^4*c^6 - 7*a^2*b^2*c^6 - 2*b^4*c^6 - a^2*c^8 + 3*b^2*c^8 - c^10) : :

Let A'B'C' be the circumcevian triangle of X(30). Let A" be the cevapoint of B' and C', and define B" and C" cyclically. AA", BB", CC" concur in X(14911). (Randy Hutson, November 2, 2017)

X(14911) lies on the cubic K528 and these lines: {30, 2986}, {186, 476}, {376, 1138}, {1294, 10420}, {2071, 3260}

X(14911) = reflection of X(2986) in X(10419)
X(14911) = perspector of circumcevian triangle of X(30) and cross-triangle of ABC and circumcevian triangle of X(30)
X(14911) = cevapoint of X(30) and (TCC-perspector of X(30))
X(14911) = barycentric product X(146)*X(2986)
X(14911) = barycentric quotient X(146)/X(3580)

X(14912) =  X(3)X(193)∩X(4)X(6)

Barycentrics    -5*a^6 + 7*a^4*b^2 - 3*a^2*b^4 + b^6 + 7*a^4*c^2 + 6*a^2*b^2*c^2 - b^4*c^2 - 3*a^2*c^4 - b^2*c^4 + c^6 : :
Barycentrics    S^2(2 SA - SW) - SA SW (SB + SC) : :
X(14912) = X(4) - 4 X(6) = X(69) - 4 X(182) = 2 X(3) + X(193) = 2 X(69) - 5 X(631) = 8 X(182) - 5 X(631) = X(20) + 2 X(1351) = 4 X(575) - X(1352) = X(193) - 4 X(1353) = X(3) + 2 X(1353) = X(376) + 2 X(1992) = 16 X(575) - 7 X(3090) = 4 X(1352) - 7 X(3090) = 8 X(141) - 11 X(3525) = 4 X(1350) - 7 X(3528) = 8 X(576) + X(3529) = 2 X(1843) - 5 X(3567) = 7 X(3090) - 10 X(3618) = 8 X(575) - 5 X(3618) = 2 X(1352) - 5 X(3618) = 17 X(3533) - 14 X(3619) = 8 X(140) - 5 X(3620) = X(1350) + 2 X(3629) = 7 X(3528) + 8 X(3629) = X(944) + 2 X(3751) = 8 X(3818) - 11 X(3855) = 16 X(3589) - 13 X(5067) = 8 X(597) - 5 X(5071) = 3 X(3524) - 4 X(5085) = 3 X(5032) - 2 X(5093) = X(74) + 2 X(5095) = X(98) + 2 X(5477) = 5 X(4) - 8 X(5480) = 5 X(6) - 2 X(5480) = 4 X(5) - X(5921) = 4 X(389) - X(6403) = 2 X(389) + X(6467) = X(6403) + 2 X(6467) = 2 X(6) + X(6776) = X(4) + 2 X(6776) = 4 X(5480) + 5 X(6776) = 2 X(1205) + X(7731) = X(5596) + 2 X(8549) = X(6776) - 4 X(8550) = X(6) + 2 X(8550) = X(5480) + 5 X(8550) = X(4) + 8 X(8550) = X(5889) + 2 X(9967) = 16 X(5092) - 13 X(10299)

X(14912) lies on the cubic K907 and these lines: {2, 3167}, {3, 193}, {4, 6}, {5, 5921}, {20, 1351}, {30, 5032}, {51, 7714}, {52, 12220}, {54, 69}, {66, 13472}, {68, 11232}, {74, 5095}, {98, 5034}, {140, 3620}, {141, 3525}, {155, 6804}, {159, 3518}, {184, 6353}, {371, 12256}, {372, 12257}, {376, 511}, {389, 6403}, {391, 6998}, {436, 14361}, {518, 7967}, {524, 3524}, {542, 3545}, {549, 11160}, {569, 11411}, {575, 1352}, {576, 3529}, {597, 5071}, {611, 1056}, {613, 1058}, {639, 6280}, {640, 6279}, {895, 12383}, {944, 3751}, {966, 7410}, {1205, 7731}, {1285, 11676}, {1350, 3528}, {1370, 1994}, {1386, 10595}, {1513, 5304}, {1570, 2549}, {1614, 1974}, {1692, 7735}, {1843, 3567}, {1899, 8889}, {1993, 7386}, {2456, 7774}, {2782, 14033}, {2794, 7739}, {3088, 11426}, {3091, 7920}, {3146, 11482}, {3168, 6618}, {3398, 14001}, {3431, 5486}, {3523, 7906}, {3533, 3619}, {3547, 13292}, {3589, 5067}, {3818, 3855}, {3839, 14848}, {5007, 8721}, {5012, 6515}, {5028, 7738}, {5039, 10788}, {5052, 11257}, {5092, 10299}, {5102, 8584}, {5135, 6977}, {5138, 6935}, {5306, 9752}, {5309, 7694}, {5422, 7392}, {5644, 10128}, {5657, 5847}, {5667, 10762}, {5820, 6879}, {5874, 11313}, {5875, 11314}, {5889, 9967}, {6108, 6770}, {6109, 6773}, {6193, 6803}, {6241, 12294}, {6337, 13335}, {6459, 8982}, {6643, 12161}, {6811, 7585}, {6813, 7586}, {6815, 8548}, {6995, 9777}, {7487, 11432}, {7493, 11003}, {7710, 9753}, {7947, 10303}, {8164, 12588}, {8593, 12243}, {9143, 12099}, {9300, 9756}, {9862, 10753}, {9924, 11431}, {9936, 11487}, {10601, 14826}, {10752, 12244}, {10754, 13172}, {10755, 13199}, {10759, 12248}, {10765, 14654}, {10766, 13200}, {11061, 11579}, {11412, 11574}, {11424, 12324}, {12176, 12177}, {12251, 13354}, {13674, 13831}, {13794, 13832}

X(14912) = midpoint of X(6776) and X(14853)
X(14912) = reflection of X(i) in X(j) for these {i,j}: {2, 5050}, {4, 14853}, {3839, 14848}, {5102, 8584}, {10516, 597}, {10519, 5085}, {11180, 10516}, {14853, 6}
X(14912) = isogonal conjugate of X(14489)
X(14912) = anticomplement of anticomplement of X(38110)
X(14912) = crosssum of X(3) and X(5093)
X(14912) = X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 1353, 193), (6, 6776, 4), (6, 8550, 6776), (51, 11206, 7714), (69, 182, 631), (140, 11898, 3620), (184, 11433, 6353), (389, 6467, 6403), (575, 1352, 3618), (597, 11180, 5071), (1352, 3618, 3090), (1587, 1588, 5254), (1899, 11427, 8889), (1899, 13366, 11427), (1992, 11179, 376), (3567, 12283, 1843), (5012, 6515, 7494), (5085, 10519, 3524), (7581, 10784, 4), (7582, 10783, 4), (8550, 12007, 6), (11061, 11579, 12317), (11245, 11402, 2)
X(14912) = second-Lemoine-circle-inverse of X(13509)
X(14912) = barycentric quotient X(6)/X(14489)

X(14913) =  X(4)X(69)∩X(51)X(193)

Barycentrics    a^2*(a^4*b^2 - b^6 + a^4*c^2 - 2*a^2*b^2*c^2 + 3*b^4*c^2 + 3*b^2*c^4 - c^6) : :
X(14913) = 3 X(51) - X(193) = 3 X(599) - X(3313) = 4 X(141) - 3 X(3819) = 5 X(3620) - 3 X(3917) = 2 X(6) - 3 X(5943) = 5 X(3618) - 6 X(6688) = 3 X(5943) - 4 X(9822) = 3 X(599) + X(9973) = X(1843) - 3 X(11188) = X(69) + 3 X(11188) = 3 X(3819) - 2 X(11574) = 3 X(5050) - 4 X(11695) = 5 X(3620) - X(12220) = 3 X(3917) - X(12220) = 3 X(2) + X(12272) = 5 X(631) - X(12283) = 3 X(10519) - 2 X(13348)

Let A'B'C' be the midheight triangle. Let A" be the trilinear pole, wrt A'B'C', of line BC, and define B" and C" cyclically. The lines A'A", B'B", C'C" concur in X(5). Let A* be the trilinear pole, wrt A'B'C', of line B"C", and define B* and C* cyclically. The lines A'A*, B'B*, C'C* concur in X(5907). Let A** be the trilinear pole, wrt A'B'C', of line B*C*, and define B** and C** cyclically. The lines A'A**, B'B**, C'C** concur in X(14913). (Randy Hutson, November 2, 2017)

X(14913) lies on the cubic K907 and these lines: {2, 6467}, {3, 9924}, {4, 69}, {6, 1196}, {51, 193}, {52, 11898}, {141, 1368}, {185, 5921}, {206, 8542}, {389, 3564}, {394, 12167}, {512, 13232}, {524, 9969}, {542, 974}, {570, 1634}, {599, 3313}, {631, 12283}, {800, 11328}, {973, 5965}, {1351, 10110}, {1353, 5462}, {2386, 3734}, {2854, 3589}, {2916, 12367}, {3618, 6688}, {3620, 3917}, {3629, 9027}, {3631, 9019}, {3964, 9737}, {5050, 11695}, {6776, 9729}, {9967, 11793}, {9976, 10821}, {10519, 13348}

X(14913) = midpoint of X(i) and X(j) for these {i,j}: {52, 11898}, {69, 1843}, {185, 5921}, {3313, 9973}, {5562, 6403}, {6467, 12272}
X(14913) = reflection of X(i) in X(j) for these {i,j}: {6, 9822}, {1351, 10110}, {1353, 5462}, {5907, 1352}, {6776, 9729}, {9967, 11793}, {11574, 141}
X(14913) = complement X(6467)

X(14913) = complement of the isotomic conjugate of X(683)
X(14913) = isotomic coinjugate of the isogonal conjugate of X(11326)
X(14913) = X(683)-complementary conjugate of X(2887)
X(14913) = crosspoint of X(2) and X(683)
X(14913) = crossdifference of every pair of points on line {3049, 3566}
X(14913) = crosssum of X(6) and X(682)
X(14913) = midheight-isotomic conjugate of X(389)
X(14913) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 12272, 6467), (6, 9822, 5943), (69, 11188, 1843), (141, 11574, 3819), (599, 9973, 3313), (3620, 12220, 3917), (5020, 6391, 6)
X(14913) = barycentric product X(76)*X(11326)

X(14914) =  X(4)X(193)∩X(51)X(3167)

Barycentrics    a^2*(a^2 - b^2 - c^2)*(3*a^8 - 12*a^6*b^2 + 6*a^4*b^4 + 12*a^2*b^6 - 9*b^8 - 12*a^6*c^2 + 16*a^4*b^2*c^2 - 20*a^2*b^4*c^2 + 32*b^6*c^2 + 6*a^4*c^4 - 20*a^2*b^2*c^4 - 46*b^4*c^4 + 12*a^2*c^6 + 32*b^2*c^6 - 9*c^8) : :
X(14914) = 2 X(1351) + X(6391) = 4 X(1351) - X(12164) = 2 X(6391) + X(12164) = 2 X(193) + X(12429)

X(14914) lies on the cubic K907 and these lines: {4, 193}, {51, 3167}, {511, 10606}, {3527, 5654}, {5102, 8681}, {6677, 11427}

X(14914) = reflection of X(3167) in X(5093)
X(14914) = {X(1351),X(6391)}-harmonic conjugate of X(12164)

X(14915) =  X(2)X(11455)∩X(23)X(74)

Barycentrics    a^2*(a^6*b^2 - 3*a^4*b^4 + 3*a^2*b^6 - b^8 + a^6*c^2 + 8*a^4*b^2*c^2 - 5*a^2*b^4*c^2 - 4*b^6*c^2 - 3*a^4*c^4 - 5*a^2*b^2*c^4 + 10*b^4*c^4 + 3*a^2*c^6 - 4*b^2*c^6 - c^8) : :

X(14915) lies on the cubic K903 and these lines: {2, 11455}, {3, 1495}, {4, 4846}, {5, 13474}, {20, 1216}, {23, 74}, {26, 3357}, {30, 511}, {51, 3830}, {52, 3146}, {64, 7387}, {68, 12324}, {110, 841}, {113, 858}, {125, 1533}, {143, 12002}, {146, 5189}, {155, 12315}, {185, 382}, {186, 13445}, {265, 10293}, {323, 14094}, {373, 381}, {376, 5891}, {378, 6800}, {389, 3627}, {399, 3292}, {468, 6699}, {500, 1480}, {546, 9729}, {547, 12045}, {548, 11793}, {550, 5447}, {631, 11439}, {1147, 1498}, {1204, 7517}, {1350, 12367}, {1514, 10297}, {1531, 7574}, {1597, 5050}, {1614, 12086}, {1657, 5562}, {2071, 14157}, {2883, 5448}, {2979, 11001}, {3024, 7286}, {3028, 5160}, {3304, 6580}, {3529, 10625}, {3534, 3917}, {3543, 5890}, {3581, 10620}, {3819, 8703}, {3839, 14845}, {3845, 5943}, {3850, 11695}, {3853, 10110}, {3854, 11465}, {3861, 12006}, {5012, 13596}, {5059, 11412}, {5066, 6688}, {5085, 9818}, {5159, 12900}, {5318, 11624}, {5321, 11626}, {5449, 6247}, {5654, 5656}, {5655, 13857}, {5878, 14790}, {6102, 13598}, {6146, 12897}, {6696, 13383}, {6759, 12038}, {7530, 11438}, {7556, 11454}, {7575, 12041}, {8718, 14118}, {9645, 10076}, {9927, 14216}, {9934, 12901}, {10095, 12102}, {10109, 10219}, {10282, 11250}, {10295, 12292}, {10296, 10721}, {10539, 11413}, {10606, 14070}, {10706, 10989}, {11440, 12088}, {11456, 13352}, {11557, 13202}, {11591, 12103}, {11806, 12295}, {12163, 13093}, {13364, 13570}

X(14915) = crossdifference of every pair of points on line {6, 9209}
X(14915) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 3426, 11472), (3, 11472, 4550), (4, 4846, 7706), (4, 12279, 10575), (20, 12162, 1216), (20, 12290, 12162), (64, 7387, 7689), (110, 7464, 10564), (125, 1533, 11799), (185, 382, 5446), (550, 5907, 5447), (1498, 12085, 1147), (3146, 6241, 52), (3426, 11820, 3), (3529, 12111, 10625), (3627, 13491, 389), (3853, 13630, 10110), (4550, 8717, 3), (6759, 12084, 12038), (7464, 12112, 110), (7574, 7728, 1531), (11472, 11820, 8717), (11591, 12103, 13348)

X(14916) =  ANTICOMPLEMENT OF X(9169)

Barycentrics    5*a^6 - 9*a^4*b^2 + 3*a^2*b^4 - b^6 - 9*a^4*c^2 + 15*a^2*b^2*c^2 - 3*b^4*c^2 + 3*a^2*c^4 - 3*b^2*c^4 - c^6 : :
X(14916) = 4 X(5108) - X(6792) = 3 X(6792) - 4 X(9169) = 3 X(5108) - X(9169)

X(14916) lies on the cubic K893 and these lines: {2, 6}, {3, 14214}, {110, 2482}, {353, 7618}, {376, 1499}, {542, 10717}, {543, 9146}, {671, 4563}, {843, 6082}, {2709, 2770}, {2721, 2753}, {4576, 8591}, {7810, 8561}, {9172, 14645}, {9830, 14360}, {10488, 14683}

X(14916) = anticomplement X(9169)
X(14916) = reflection of X(i) in X(j) for these {i,j}: {2, 5108}, {6792, 2}, {10717, 12036}
X(14916) = anticomplement X(9169)
X(14916) = {X(599),X(1641)}-harmonic conjugate of X(2)
X(14916) = orthoptic-circle-of-Steiner-circumellipe-inverse of X(7840)
X(14916) = psi-transform of X(2482)
X(14916) = X(3448)-of-McCay-triangle

X(14917) =  X(30)X(511)∩X(39)X(184)

Barycentrics    a^2*(a^6*b^2 - a^4*b^4 + a^2*b^6 - b^8 + a^6*c^2 - a^2*b^4*c^2 - a^4*c^4 - a^2*b^2*c^4 + 2*b^4*c^4 + a^2*c^6 - c^8) : :

X(14917) lies on the cubic K890 and these lines: {30, 511}, {39, 184}, {51, 5309}, {76, 11442}, {186, 11653}, {2021, 3455}, {3819, 7880}, {3917, 7801}, {5052, 8541}, {5943, 7817}, {8779, 14574}, {14569, 14715}

X(14918) =  ISOGONAL CONJUGATE OF X(11077)

Barycentrics    (a^2 + b^2 - c^2)*(a^2 - b^2 - b*c - c^2)*(a^2 - b^2 + b*c - c^2)*(a^2 - b^2 + c^2)*(a^2*b^2 - b^4 + a^2*c^2 + 2*b^2*c^2 - c^4) : :

X(14918) lies on the cubic K856 and these lines: {2, 95}, {4, 1209}, {20, 10600}, {53, 311}, {94, 9381}, {128, 186}, {297, 525}, {323, 340}, {445, 3219}, {470, 10633}, {471, 10632}, {1249, 6515}, {1273, 11062}, {1629, 3410}, {2052, 11140}, {2888, 8884}, {3078, 10184}, {11078, 11093}, {11092, 11094}

X(14918) = isogonal conjugate of X(11077)
X(14918) = isotomic conjugate of the isogonal conjugate of X(11062)
X(14918) = X(14918) = X(i)-complementary conjugate of X(j) for these (i,j): {1953, 113}, {2159, 140}, {2179, 3163}, {2349, 3819}
X(14918) = X(340)-Ceva conjugate of X(1154)
X(14918) = X(1154)-cross conjugate of X(1273)
X(14918) = X(i)-isoconjugate of X(j) for these (i,j): {1, 11077}, {48, 1141}, {265, 2148}, {1989, 2169}, {2166, 14533}, {2168, 5961}
X(14918) = cevapoint of X(1154) and X(11062)
X(14918) = polar conjugate of X(1141)
X(14918) = pole wrt polar circle of trilinear polar of X(1141) (line X(6)X(2623))
X(14918) = barycentric product X(i)*X(j) for these {i,j}: {4, 1273}, {5, 340}, {53, 7799}, {76, 11062}, {186, 311}, {264, 1154}, {298, 6116}, {299, 6117}, {323, 324}, {343, 14165}, {1969, 2290}, {2081, 6331}
X(14918) = barycentric quotient X(i)/X(j) for these {i,j}: {4, 1141}, {5, 265}, {6, 11077}, {50, 14533}, {52, 5961}, {53, 1989}, {128, 539}, {186, 54}, {311, 328}, {323, 97}, {324, 94}, {340, 95}, {562, 252}, {1154, 3}, {1273, 69}, {2081, 647}, {2290, 48}, {2914, 1157}, {3199, 11060}, {5962, 96}, {6116, 13}, {6117, 14}, {6149, 2169}, {6344, 14859}, {11062, 6}, {11587, 3484}, {12077, 14582}, {13450, 6344}, {14165, 275}, {14591, 14586}
X(14918) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (324, 467, 14129), (340, 14165, 323), (343, 467, 324)

X(14919) =  ISOGONAL CONJUGATE OF X(1990)

Barycentrics    (cos A)/(cos A - 2 cos B cos C) : :
Barycentrics    a^2*(a^2 - b^2 - c^2)*(a^4 - 2*a^2*b^2 + b^4 + a^2*c^2 + b^2*c^2 - 2*c^4)*(a^4 + a^2*b^2 - 2*b^4 - 2*a^2*c^2 + b^2*c^2 + c^4) : :

X(14919) lies on the MacBeath circumconic, the cubics K564 and K856, and these lines: {2, 648}, {3, 74}, {23, 1297}, {97, 6509}, {122, 3448}, {276, 4993}, {287, 14417}, {297, 10718}, {323, 3284}, {394, 4558}, {647, 10766}, {651, 1214}, {684, 895}, {1073, 1993}, {1217, 3091}, {1331, 3682}, {1332, 3998}, {1636, 8552}, {1650, 9140}, {2072, 5627}, {2132, 10733}, {2394, 2986}, {2433, 2987}, {2967, 11284}, {3146, 3346}, {3153, 10421}, {3926, 4563}, {5159, 12079}, {5468, 6394}, {5481, 7496}, {8798, 12086}, {10752, 14685}, {11064, 11079}

X(14919) = isogonal conjugate of X(1990)
X(14919) = X(1494)-Ceva conjugate of X(74)
X(14919) = X(i)-cross conjugate of X(j) for these (i,j): {323, 97}, {1636, 110}, {2430, 6080}, {3284, 3}, {8552, 4558}, {13754, 69}
X(14919) = cevapoint of X(i) and X(j) for these (i,j): {3, 3284}, {6, 6000}, {1636, 2972}
X(14919) = trilinear pole of line X(3)X(520)
X(14919) = crossdifference of every pair of points on line {1637, 9409}
X(14919) = crosssum of X(1495) and X(14581)
X(14919) = X(i)-zayin conjugate of X(j) for these (i,j): {1, 1990}, {46, 2173}
X(14919) = X(i)-isoconjugate of X(j) for these (i,j): {1, 1990}, {4, 2173}, {6, 1784}, {19, 30}, {25, 14206}, {33, 6357}, {34, 7359}, {75, 14581}, {92, 1495}, {107, 2631}, {108, 14400}, {158, 3284}, {162, 1637}, {264, 9406}, {661, 4240}, {811, 14398}, {823, 9409}, {1096, 11064}, {1099, 8749}, {1783, 11125}, {1897, 14399}, {1969, 9407}, {1973, 3260}, {2153, 6110}, {2154, 6111}, {6793, 8767}
X(14919) = barycentric product X(i)*X(j) for these {i,j}: {3, 1494}, {63, 2349}, {69, 74}, {99, 14380}, {304, 2159}, {328, 14385}, {1304, 3265}, {2394, 4558}, {2433, 4563}, {3926, 8749}, {6390, 9139}, {7799, 11079}
X(14919) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 1784}, {3, 30}, {6, 1990}, {15, 6110}, {16, 6111}, {32, 14581}, {48, 2173}, {63, 14206}, {69, 3260}, {74, 4}, {110, 4240}, {184, 1495}, {219, 7359}, {222, 6357}, {265, 14254}, {394, 11064}, {520, 9033}, {577, 3284}, {647, 1637}, {652, 14400}, {822, 2631}, {895, 9214}, {1304, 107}, {1459, 11125}, {1494, 264}, {1636, 14401}, {2159, 19}, {2349, 92}, {2394, 14618}, {2433, 2501}, {2972, 1650}, {3049, 14398}, {3284, 3163}, {3292, 5642}, {4558, 2407}, {5562, 1568}, {5627, 6344}, {5663, 11251}, {6000, 133}, {8552, 5664}, {8749, 393}, {8779, 6793}, {9247, 9406}, {9717, 468}, {10152, 14249}, {10419, 1300}, {10605, 1514}, {11079, 1989}, {12079, 2970}, {12096, 3184}, {13754, 113}, {14264, 403}, {14379, 11589}, {14380, 523}, {14385, 186}, {14575, 9407}

X(14920) =  ISOGONAL CONJUGATE OF X(11079)

Barycentrics    (csc 2A)(sin 3A)(cos A - 2 cos B cos C) : :
Barycentrics    (a^2 + b^2 - c^2)*(a^2 - b^2 - b*c - c^2)*(a^2 - b^2 + b*c - c^2)*(a^2 - b^2 + c^2)*(2*a^4 - a^2*b^2 - b^4 - a^2*c^2 + 2*b^2*c^2 - c^4) : :

X(14920) lies on the cubic K856 and these lines: {2, 648}, {4, 110}, {94, 264}, {140, 2972}, {275, 13582}, {297, 9141}, {323, 340}, {468, 2967}, {470, 11092}, {471, 11078}, {877, 7664}, {1235, 14389}, {1990, 3260}, {3184, 3522}, {3268, 5664}, {4240, 5642}, {7879, 8743}, {9979, 14401}

X(14920) = isogonal conjugate of X(11079)
X(14920) = isotomic conjugate of isogonal conjugate of X(39176)
X(14920) = crosspoint of X(470) and X(471)
X(14920) = X(i)-complementary conjugate of X(j) for these (i,j): {2148, 30}, {2173, 1209}, {9406, 233}
X(14920) = X(264)-Ceva conjugate of X(30)
X(14920) = X(1511)-cross conjugate of X(6148)
X(14920) = cevapoint of X(113) and X(3163)
X(14920) = crosspoint of X(470) and X(471)
X(14920) = crosssum of X(36296) and X(36297)
X(14920) = crossdifference of every pair of points on line {686, 9409}
X(14920) = pole wrt polar circle of trilinear polar of X(5627) (line X(1637)X(1989))
X(14920) = polar conjugate of X(5627)
X(14920) = X(i)-isoconjugate of X(j) for these (i,j): {1, 11079}, {48, 5627}, {265, 2159}
X(14920) = barycentric product X(i)*X(j) for these {i,j}: {4, 6148}, {30, 340}, {186, 3260}, {264, 1511}, {298, 6111}, {299, 6110}, {648, 5664}, {1990, 7799}, {3268, 4240}, {11064, 14165}
X(14920) = barycentric quotient X(i)/X(j) for these {i,j}: {4, 5627}, {6, 11079}, {30, 265}, {186, 74}, {340, 1494}, {526, 14380}, {1511, 3}, {1637, 14582}, {1784, 2166}, {1986, 14264}, {1990, 1989}, {2914, 3470}, {3043, 14385}, {3258, 125}, {3260, 328}, {4240, 476}, {5664, 525}, {6110, 14}, {6111, 13}, {6148, 69}, {14581, 11060}, {14583, 14595}

X(14921) =  ISOGONAL CONJUGATE OF X(11089)

Barycentrics    (a^2 - b^2 - b*c - c^2)*(a^2 - b^2 + b*c - c^2)*(Sqrt[3]*a^2 - 2*S) : :

X(14921) lies on the cubic K856 and these lines: {2, 6151}, {298, 6148}, {299, 1273}, {302, 16537}, {323, 7799}, {533, 30459}, {616, 33498}, {618, 38993}, {3268, 6138}, {11127, 30471}, {11130, 30472}

X(14921) = isogonal conjugate of X(11089)
X(14921) = X(i)-Ceva conjugate of X(j) for these (i,j): {298, 533}, {11128, 6148}, {11133, 1273}
X(14921) = crosspoint of X(298) and X(7799)
X(14921) = crosssum of X(3457) and X(11060)
X(14921) = X(298)-daleth conjugate of X(6148)
X(14921) = X(i)-isoconjugate of X(j) for these (i,j): {1, 11089}, {2154, 2381}
X(14921) = barycentric product X(i)*X(j) for these {i,j}: {298, 619}, {299, 533}, {395, 7799}
X(14921) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 11089}, {16, 2381}, {298, 11120}, {299, 11118}, {323, 6151}, {395, 1989}, {533, 14}, {619, 13}, {3480, 14372}, {10411, 10410}, {14368, 1338}

X(14922) =  ISOGONAL CONJUGATE OF X(11084)

Barycentrics    (a^2 - b^2 - b*c - c^2)*(a^2 - b^2 + b*c - c^2)*(Sqrt(3)*a^2 + 2*S) : :

X(14922) lies on the cubic K856 and these lines: {2, 2981}, {298, 1273}, {299, 6148}, {323, 7799}, {3268, 6137}

X(14922) = isogonal conjugate of X(11084)
X(14922) = X(i)-Ceva conjugate of X(j) for these (i,j): {299, 532}, {11129, 6148}, {11132, 1273}
X(14922) = crosspoint of X(299) and X(7799)
X(14922) = crosssum of X(3458) and X(11060)
X(14922) = X(299)-daleth conjugate of X(6148)
X(14922) = X(i)-isoconjugate of X(j) for these (i,j): {1, 11084}, {2153, 2380}
X(14922) = barycentric product X(i)*X(j) for these {i,j}: {298, 532}, {299, 618}, {396, 7799}
X(14922) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 11084}, {15, 2380}, {298, 11117}, {299, 11119}, {323, 2981}, {396, 1989}, {532, 13}, {618, 14}, {3479, 14373}, {10411, 10409}, {14369, 1337}

X(14923) =  X(1)X(88)∩X(4)X(8)

Barycentrics    a*(a^2*b - b^3 + a^2*c - 3*a*b*c + 2*b^2*c + 2*b*c^2 - c^3) : :
X(14923) = 3 X(8) - 2 X(72) = 4 X(942) - 3 X(3241) = 4 X(960) - 5 X(3617) = 6 X(354) - 5 X(3623) = 8 X(72) - 9 X(3681) = 4 X(8) - 3 X(3681) = 9 X(2) - 10 X(3698) = 3 X(3057) - 5 X(3698) = 5 X(3616) - 6 X(3753) = 3 X(1) - 4 X(3754) = 7 X(3622) - 8 X(3812) = 4 X(72) - 3 X(3869) = 3 X(3681) - 2 X(3869) = 4 X(65) - 3 X(3873) = 2 X(145) - 3 X(3873) = 6 X(3679) - 5 X(3876) = 4 X(10) - 3 X(3877) = 5 X(3876) - 4 X(3878) = 3 X(3679) - 2 X(3878) = 5 X(1698) - 4 X(3884) = 8 X(3754) - 3 X(3885) = 4 X(3244) - 5 X(3889) = 6 X(2) - 5 X(3890) = 4 X(3057) - 5 X(3890) = 4 X(3698) - 3 X(3890) = 7 X(3624) - 6 X(3898) = 4 X(3678) - 3 X(3899) = 7 X(3624) - 8 X(3918) = 3 X(3898) - 4 X(3918) = 2 X(3635) - 3 X(3919) = 6 X(3742) - 7 X(3922) = 15 X(3617) - 14 X(3983) = 6 X(960) - 7 X(3983) = 15 X(3876) - 16 X(4015) = 9 X(3679) - 8 X(4015)

X(14923) lies on these lines: {1, 88}, {2, 3057}, {3, 4861}, {4, 8}, {5, 1145}, {7, 13601}, {9, 7673}, {10, 3877}, {11, 8256}, {12, 10129}, {21, 5119}, {35, 3897}, {40, 2975}, {56, 8668}, {57, 1476}, {63, 4853}, {65, 145}, {78, 6915}, {149, 1837}, {165, 5303}, {169, 644}, {200, 11531}, {210, 4678}, {224, 13375}, {226, 12640}, {346, 2262}, {354, 3623}, {388, 7702}, {392, 9780}, {443, 1000}, {484, 8666}, {497, 5554}, {515, 9961}, {518, 1278}, {519, 3868}, {528, 10394}, {643, 11101}, {672, 4051}, {693, 9366}, {758, 3632}, {908, 4301}, {942, 3241}, {944, 6948}, {946, 6735}, {956, 12702}, {960, 3617}, {978, 4695}, {993, 11010}, {995, 3987}, {997, 5330}, {1122, 4373}, {1155, 11260}, {1319, 4188}, {1376, 2098}, {1387, 13747}, {1388, 4881}, {1482, 4511}, {1483, 10609}, {1616, 7292}, {1621, 1697}, {1698, 3884}, {1706, 7962}, {1788, 10529}, {2093, 12629}, {2099, 3913}, {2136, 3340}, {2170, 3501}, {2171, 3169}, {2475, 3909}, {2476, 10039}, {2800, 5881}, {3146, 12125}, {3174, 11526}, {3218, 12513}, {3244, 3889}, {3245, 5288}, {3485, 10528}, {3487, 11239}, {3496, 4390}, {3616, 3753}, {3622, 3812}, {3624, 3898}, {3625, 5904}, {3626, 5692}, {3633, 3874}, {3635, 3919}, {3655, 13145}, {3660, 6049}, {3678, 3899}, {3679, 3876}, {3742, 3922}, {3894, 4757}, {3902, 10449}, {3930, 4050}, {3935, 12635}, {3962, 4661}, {3984, 4882}, {4004, 5045}, {4005, 4711}, {4067, 4701}, {4134, 4746}, {4342, 8582}, {4420, 5730}, {4499, 9519}, {4915, 12526}, {4919, 9310}, {5046, 12701}, {5123, 5154}, {5250, 5260}, {5284, 9819}, {5440, 10222}, {5552, 5603}, {5657, 6891}, {5690, 6882}, {5722, 9802}, {5734, 13600}, {5748, 5806}, {5818, 6973}, {5853, 7672}, {5854, 10944}, {5886, 10284}, {6734, 6943}, {6796, 11014}, {6797, 11373}, {6890, 14110}, {6953, 7080}, {7308, 11530}, {7701, 11525}, {8583, 9342}, {9581, 10707}, {10826, 12758}, {10915, 12047}

X(14923) = reflection of X(i) in X(j) for these {i,j}: {8, 10914}, {145, 65}, {3057, 5836}, {3621, 3893}, {3633, 3874}, {3868, 5903}, {3869, 8}, {3885, 1}, {4067, 4701}, {5697, 10}, {5904, 3625}, {7673, 9}, {12528, 5881}
X(14923) = anticomplement X(3057)
X(14923) = Fuhrmann-circle-inverse of X(5176)
X(14923) = anticomplement of the isogonal of X(1476)
X(14923) = X(8)-beth conjugate of X(5697)
X(14923) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {1222, 3436}, {1476, 8}, {3451, 2}
X(14923) = crosspoint of X(668) and X(5382)
X(14923) = X(20)-of-inner-Conway-triangle
X(14923) = X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 4642, 4850), (1, 5541, 8715), (2, 3057, 3890), (4, 8, 5176), (8, 962, 3436), (8, 3434, 5086), (8, 3869, 3681), (8, 5082, 5178), (8, 11415, 3421), (10, 5697, 3877), (10, 7741, 7705), (40, 3872, 2975), (65, 145, 3873), (145, 4190, 3476), (200, 11531, 11682), (404, 1320, 1), (946, 6735, 11681), (962, 3436, 5057), (1482, 5687, 4511), (2136, 3340, 3870), (3057, 5836, 2), (3244, 5902, 3889), (3679, 3878, 3876), (3812, 5919, 3622), (3898, 3918, 3624), (3899, 4668, 3678), (4301, 6736, 908), (4853, 7991, 63), (5082, 12245, 8), (5250, 9623, 5260), (8256, 13463, 11)
X(14923) = barycentric product X(i)*X(j) for these {i,j}: {321, 7419}, {5382, 5510}
X(14923) = barycentric quotient X(i)/X(j) for these {i,j}: {7419, 81}, {14261, 8056}

X(14924) =  X(2)X(11477)∩X(154)X(10541)

Barycentrics    a^2*(a^4 - 6*a^2*b^2 + 5*b^4 - 6*a^2*c^2 - 74*b^2*c^2 + 5*c^4) : :

X(14924) lies on the Thomson-Gibert-Moses hyperbola and these lines: {2, 11477}, {154, 10541}, {373, 5646}, {392, 3646}, {575, 3167}, {576, 5544}, {1201, 2334}, {1350, 5888}, {1656, 5655}, {1995, 6030}, {3090, 5656}, {3628, 5654}, {5085, 7712}, {5644, 11482}

X(14925) =  X(3)X(64)∩X(40)X(10535)

Barycentrics    a^2*(a^11 - a^10*b - 4*a^9*b^2 + 4*a^8*b^3 + 6*a^7*b^4 - 6*a^6*b^5 - 4*a^5*b^6 + 4*a^4*b^7 + a^3*b^8 - a^2*b^9 - a^10*c + 4*a^9*b*c + 2*a^8*b^2*c - 10*a^7*b^3*c + 6*a^5*b^5*c - 2*a^4*b^6*c + 2*a^3*b^7*c + a^2*b^8*c - 2*a*b^9*c - 4*a^9*c^2 + 2*a^8*b*c^2 + 6*a^7*b^2*c^2 - 2*a^5*b^4*c^2 - 4*a^4*b^5*c^2 + 2*a^3*b^6*c^2 - 2*a*b^8*c^2 + 2*b^9*c^2 + 4*a^8*c^3 - 10*a^7*b*c^3 + 8*a^5*b^3*c^3 + 2*a^4*b^4*c^3 - 2*a^3*b^5*c^3 - 4*a^2*b^6*c^3 + 4*a*b^7*c^3 - 2*b^8*c^3 + 6*a^7*c^4 - 2*a^5*b^2*c^4 + 2*a^4*b^3*c^4 - 6*a^3*b^4*c^4 + 4*a^2*b^5*c^4 + 2*a*b^6*c^4 - 6*b^7*c^4 - 6*a^6*c^5 + 6*a^5*b*c^5 - 4*a^4*b^2*c^5 - 2*a^3*b^3*c^5 + 4*a^2*b^4*c^5 - 4*a*b^5*c^5 + 6*b^6*c^5 - 4*a^5*c^6 - 2*a^4*b*c^6 + 2*a^3*b^2*c^6 - 4*a^2*b^3*c^6 + 2*a*b^4*c^6 + 6*b^5*c^6 + 4*a^4*c^7 + 2*a^3*b*c^7 + 4*a*b^3*c^7 - 6*b^4*c^7 + a^3*c^8 + a^2*b*c^8 - 2*a*b^2*c^8 - 2*b^3*c^8 - a^2*c^9 - 2*a*b*c^9 + 2*b^2*c^9) : :

X(14925) lies on the cubic K844 and these lines: {3, 64}, {40, 10535}, {56, 945}, {184, 7412}, {578, 2194}, {1503, 6922}, {2077, 6285}, {2192, 10306}, {3362, 6056}, {5878, 6948}, {6827, 9833}, {6865, 11206}, {6891, 14216}, {10268, 10536}

X(14926) =  REFLECTION OF X(3) IN X(5888)

Barycentrics    a^2*(a^8 - 6*a^4*b^4 + 8*a^2*b^6 - 3*b^8 - 3*a^4*b^2*c^2 + 17*a^2*b^4*c^2 - 14*b^6*c^2 - 6*a^4*c^4 + 17*a^2*b^2*c^4 + 34*b^4*c^4 + 8*a^2*c^6 - 14*b^2*c^6 - 3*c^8) : :
X(14926) = 5 X(5888) + X(13603) = 5 X(3) + 2 X(13603) = 7 X(3526) - 2 X(13623)

X(14926) lies on these lines: {2, 10620}, {3, 5888}, {5, 7693}, {64, 3526}, {182, 399}, {195, 14128}, {381, 1350}, {1173, 11591}, {1656, 5448}, {5054, 11472}, {5055, 7699}, {5097, 5891}, {5651, 12302}, {10117, 11204}

X(14926) = reflection of X(i) in X(j) for these {i,j}: {3, 5888}, {7693, 5}

X(14927) =  X(4)X(83)∩X(20)X(64)

Barycentrics    -7*a^6 + a^4*b^2 + 3*a^2*b^4 + 3*b^6 + a^4*c^2 + 2*a^2*b^2*c^2 - 3*b^4*c^2 + 3*a^2*c^4 - 3*b^2*c^4 + 3*c^6 : :
Barycentrics    S^2 SA + 2 (2 SA a^2 - SB SC) SW : :
X(14927) = 3 X(4) - 4 X(182) = 3 X(69) - 4 X(1350) = 3 X(20) - 2 X(1350) = 3 X(376) - 2 X(1352) = 5 X(1351) - 6 X(1353) = 16 X(1353) - 15 X(1992) = 8 X(1351) - 9 X(1992) = 4 X(141) - 5 X(3522) = 16 X(182) - 15 X(3618) = 4 X(4) - 5 X(3618) = 8 X(3) - 7 X(3619) = 5 X(631) - 4 X(3818) = 8 X(3589) - 7 X(3832) = 2 X(3627) - 3 X(5050) = 5 X(3091) - 6 X(5085) = 7 X(3090) - 8 X(5092) = 3 X(3543) - 4 X(5480) = 3 X(69) - 2 X(5921) = 3 X(20) - X(5921) = 4 X(1353) - 5 X(6776) = 3 X(1992) - 4 X(6776) = 2 X(1351) - 3 X(6776) = 2 X(3416) - 3 X(9778) = 4 X(1386) - 3 X(9812), 7 X(3523) - 6 X(10516), 4 X(550) - 3 X(10519), 4 X(9969) - 5 X(10574), 4 X(3529) + X(11008), 4 X(576) - X(11541), 4 X(546) - 5 X(12017), 3 X(5622) - 2 X(12295), 10 X(182) - 9 X(14561), 5 X(4) - 6 X(14561), 9 X(376) - 8 X(14810), 3 X(1352) - 4 X(14810), 2 X(382) - 3 X(14853)

X(14927) lies on the cubic K820 and these lines: {3, 3619}, {4, 83}, {6, 3146}, {20, 64}, {30, 1351}, {66, 11454}, {110, 1370}, {141, 3522}, {154, 7396}, {159, 11413}, {183, 3424}, {193, 5059}, {376, 1352}, {382, 14853}, {511, 3529}, {516, 3875}, {518, 12536}, {542, 11001}, {546, 12017}, {550, 10519}, {576, 11541}, {631, 3818}, {637, 14227}, {638, 14242}, {1007, 5999}, {1131, 13910}, {1132, 13972}, {1296, 2366}, {1386, 9812}, {1428, 5225}, {1657, 3564}, {1853, 10565}, {2330, 5229}, {2777, 11061}, {3066, 6995}, {3090, 5092}, {3091, 5085}, {3313, 12111}, {3416, 9778}, {3523, 10516}, {3534, 11180}, {3543, 5480}, {3589, 3832}, {3627, 5050}, {3796, 7378}, {3827, 9961}, {4344, 7247}, {5207, 6337}, {5596, 6225}, {5622, 12295}, {5651, 7386}, {5800, 10431}, {5870, 12322}, {5871, 12323}, {6803, 13419}, {7391, 11003}, {7408, 10601}, {7494, 11550}, {7500, 11433}, {7528, 13339}, {7667, 14826}, {9969, 10574}, {11381, 11574}, {12220, 12279}

X(14927) = midpoint of X(i) and X(j) for these {i,j}: {193, 5059}, {12220, 12279}
X(14927) = reflection of X(i) in X(j) for these {i,j}: {69, 20}, {3146, 6}, {5921, 1350}, {6225, 5596}, {11180, 3534}, {11381, 11574}, {12111, 3313}
X(14927) = anticomplement of X(36990)
X(14927) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (20, 5921, 1350), (1350, 5921, 69), (5999, 7710, 1007)

X(14928) =  X(6)X(543)∩X(69)X(74)

Barycentrics    4*a^6 - 2*a^4*b^2 - a^2*b^4 - b^6 - 2*a^4*c^2 - 2*a^2*b^2*c^2 + 3*b^4*c^2 - a^2*c^4 + 3*b^2*c^4 - c^6 : :
X(14928) = X(69) - 3 X(99) = 2 X(141) - 3 X(2482) = 3 X(115) - 4 X(3589) = 3 X(671) - 5 X(3618) = 6 X(620) - 5 X(3763) = 3 X(114) - 2 X(3818) = 2 X(3589) - 3 X(5026) = X(148) - 3 X(5182) = 2 X(3629) - 3 X(5477) = 4 X(5092) - 3 X(6055) = X(193) + 3 X(8591) = X(193) - 3 X(8593) = 5 X(3620) - 3 X(11161) = 3 X(5085) - 2 X(11623) = 5 X(3763) - 3 X(11646) = 3 X(11632) - 5 X(12017) = X(316) - 3 X(12215)

X(14928) lies on the cubic K819 and these lines: {6, 543}, {69, 74}, {110, 14360}, {114, 3818}, {115, 3589}, {125, 7664}, {126, 2502}, {141, 2482}, {148, 5182}, {159, 2936}, {182, 11185}, {193, 8591}, {316, 12215}, {325, 11645}, {511, 10992}, {524, 6781}, {620, 3763}, {671, 3618}, {1352, 9734}, {1503, 6390}, {2782, 13354}, {2794, 14532}, {3620, 11161}, {3629, 5477}, {4027, 10336}, {4048, 7820}, {4563, 10553}, {5085, 11623}, {5092, 5939}, {6144, 14645}, {7777, 8592}, {7806, 8289}, {8972, 13642}, {9143, 9146}, {10753, 13172}, {11632, 12017}, {13761, 13941}

X(14928) = midpoint of X(i) and X(j) for these {i,j}: {8591, 8593}, {10753, 13172}
X(14928) = reflection of X(i) in X(j) for these {i,j}: {115, 5026}, {11646, 620}
X(14928) = crossdifference of every pair of points on line {9023, 14398}

X(14929) =  X(2)X(3793)∩X(30)X(69)

Barycentrics    -4*a^4 + a^2*b^2 + 3*b^4 + a^2*c^2 + 2*b^2*c^2 + 3*c^4 : :
X(14929) = 3 X(599) - X(7737) = X(6144) - 3 X(7739) = 3 X(7761) - X(7798) = 3 X(141) - 2 X(7804) = X(7804) - 3 X(7848) = 2 X(3589) - 3 X(7865) = 5 X(3620) - 3 X(11286) = X(193) - 3 X(11287)

X(14929) lies on these lines: {2, 3793}, {5, 183}, {30, 69}, {76, 3627}, {99, 550}, {140, 1007}, {141, 754}, {193, 11287}, {230, 7818}, {316, 3845}, {325, 549}, {376, 10513}, {427, 1369}, {524, 7761}, {538, 3630}, {548, 3926}, {599, 7737}, {625, 13468}, {632, 1078}, {1384, 8368}, {2542, 6190}, {2543, 6189}, {2896, 3329}, {3314, 8369}, {3589, 7865}, {3620, 11286}, {3629, 4045}, {3631, 3734}, {3815, 7810}, {3858, 7860}, {3945, 11359}, {4030, 7272}, {4894, 7198}, {5024, 8358}, {5077, 11160}, {5232, 11354}, {5254, 7826}, {5305, 7784}, {5306, 7853}, {5468, 11007}, {5590, 13644}, {5591, 13763}, {5921, 14532}, {6144, 7739}, {6390, 7788}, {6656, 7766}, {6722, 7780}, {7603, 11168}, {7735, 8360}, {7745, 7854}, {7754, 8357}, {7774, 8359}, {7779, 8356}, {7789, 7896}, {7792, 7883}, {7807, 7939}, {7819, 7879}, {7830, 7882}, {7877, 7936}, {7890, 9607}, {7904, 7946}

X(14929) = midpoint of X(i) and X(j) for these {i,j}: {5077, 11160}, {5921, 14532}
X(14929) = reflection of X(i) in X(j) for these {i,j}: {141, 7848}, {3629, 4045}, {3734, 3631}
X(14929) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (315, 7767, 5), (2896, 7762, 8362), (3785, 7776, 140), (3933, 7750, 550), (7750, 7768, 3933), (7784, 14023, 5305), (7810, 7845, 3815), (7811, 7850, 325), (7826, 7873, 5254), (7893, 7929, 6656)

X(14930) =  X(2)X(6)∩X(20)X(9605)

Barycentrics    7*a^4 + 10*a^2*b^2 - b^4 + 10*a^2*c^2 + 2*b^2*c^2 - c^4 : :

X(14930) lies on these lines: {2, 6}, {20, 9605}, {30, 14482}, {39, 3522}, {145, 9575}, {251, 5065}, {393, 7409}, {439, 13356}, {1180, 13341}, {1249, 7378}, {1285, 5024}, {1503, 14484}, {1627, 13342}, {2548, 5068}, {2549, 3146}, {2999, 5838}, {3087, 7408}, {3424, 8550}, {3832, 5041}, {3926, 7878}, {5056, 5305}, {5059, 7738}, {5477, 5984}, {7000, 7581}, {7374, 7582}, {7803, 7926}, {9744, 9748}

X(14930) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6, 7736, 5304), (6, 9300, 7735), (193, 3329, 2), (1285, 5024, 10304), (5304, 7736, 2), (7585, 7586, 3618)

X(14931) =  X(2)X(99)∩X(6)X(1916)

Barycentrics    a^8 - 2*a^2*b^6 + 3*b^4*c^4 - 2*a^2*c^6 : :

X(14931) lies on these lines: {2, 99}, {6, 1916}, {98, 3098}, {187, 5152}, {385, 5104}, {542, 7779}, {690, 5987}, {1502, 9063}, {2023, 8290}, {2782, 5999}, {3311, 8310}, {3312, 8311}, {3314, 11646}, {3329, 5026}, {3972, 12191}, {6034, 7875}, {6199, 8312}, {6221, 8304}, {6321, 13862}, {6323, 9066}, {6395, 8313}, {6398, 8305}, {6468, 8306}, {6469, 8307}, {7766, 10754}, {7840, 9830}, {7898, 9878}, {8592, 11163}, {8667, 10810}, {9772, 13188}

X(14931) = reflection of X(385) in X(5939)
X(14931) = X(2)-Hirst inverse of X(3734)
X(14931) = inverse-in-Steiner-circumellipse of X(3734)
X(14931) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 9877, 10487), (6, 5989, 8289), (6, 8289, 4027), (99, 671, 3734), (671, 7919, 115), (1916, 5989, 4027), (1916, 8289, 6), (8302, 8303, 99), (13188, 13860, 9772)

X(14932) =  MIDPOINT OF X(9185) AND X(14698)

Barycentrics    (b - c)*(b + c)*(-7*a^10 + 14*a^8*b^2 - 7*a^6*b^4 + 2*a^4*b^6 - 4*a^2*b^8 + 2*b^10 + 14*a^8*c^2 - 28*a^6*b^2*c^2 + 12*a^4*b^4*c^2 + 8*a^2*b^6*c^2 - 4*b^8*c^2 - 7*a^6*c^4 + 12*a^4*b^2*c^4 - 15*a^2*b^4*c^4 + 2*b^6*c^4 + 2*a^4*c^6 + 8*a^2*b^2*c^6 + 2*b^4*c^6 - 4*a^2*c^8 - 4*b^2*c^8 + 2*c^10) : :

X(14932) lies on the Hutson-Parry circle, the cubic K733, and these lines: {2, 690}, {13, 9200}, {14, 9201}, {110, 476}, {111, 1640}, {526, 5640}, {542, 5466}, {804, 9159}, {1499, 1551}, {1637, 6792}, {1992, 9003}, {3569, 6032}, {5642, 9168}, {7698, 14420}, {8371, 9140}, {14644, 14846}

X(14932) = midpoint of X(9185) and X(14698)
X(14932) = reflection of X(i) in X(j) for these {i,j}: {9138, 9185}, {9140, 8371}, {9168, 5642}, {9180, 5465}, {9185, 14697}
X(14932) = {X(14697),X(14698)}-harmonic conjugate of X(9138)
X(14932) = crossdifference of every pair of points on line {2088, 2502}

X(14933) =  X(3)X(526)∩X(30)X(74)

Barycentrics    a^2*(a^18*b^2 - 5*a^16*b^4 + 7*a^14*b^6 + 7*a^12*b^8 - 35*a^10*b^10 + 49*a^8*b^12 - 35*a^6*b^14 + 13*a^4*b^16 - 2*a^2*b^18 + a^18*c^2 - 4*a^16*b^2*c^2 + 13*a^14*b^4*c^2 - 38*a^12*b^6*c^2 + 65*a^10*b^8*c^2 - 56*a^8*b^10*c^2 + 19*a^6*b^12*c^2 + 2*a^4*b^14*c^2 - 2*a^2*b^16*c^2 - 5*a^16*c^4 + 13*a^14*b^2*c^4 + 2*a^12*b^4*c^4 - 5*a^10*b^6*c^4 - 56*a^8*b^8*c^4 + 89*a^6*b^10*c^4 - 44*a^4*b^12*c^4 + 7*a^2*b^14*c^4 - b^16*c^4 + 7*a^14*c^6 - 38*a^12*b^2*c^6 - 5*a^10*b^4*c^6 + 104*a^8*b^6*c^6 - 71*a^6*b^8*c^6 - 14*a^4*b^10*c^6 + 11*a^2*b^12*c^6 + 6*b^14*c^6 + 7*a^12*c^8 + 65*a^10*b^2*c^8 - 56*a^8*b^4*c^8 - 71*a^6*b^6*c^8 + 86*a^4*b^8*c^8 - 14*a^2*b^10*c^8 - 15*b^12*c^8 - 35*a^10*c^10 - 56*a^8*b^2*c^10 + 89*a^6*b^4*c^10 - 14*a^4*b^6*c^10 - 14*a^2*b^8*c^10 + 20*b^10*c^10 + 49*a^8*c^12 + 19*a^6*b^2*c^12 - 44*a^4*b^4*c^12 + 11*a^2*b^6*c^12 - 15*b^8*c^12 - 35*a^6*c^14 + 2*a^4*b^2*c^14 + 7*a^2*b^4*c^14 + 6*b^6*c^14 + 13*a^4*c^16 - 2*a^2*b^2*c^16 - b^4*c^16 - 2*a^2*c^18) : :

X(14933) lies on the cubic K725 and these lines: {3, 526}, {30, 74}, {6760, 13557}

X(14934) =  MIDPOINT OF X(110) AND X(477)

Barycentrics    2*a^16 - 6*a^14*b^2 + a^12*b^4 + 15*a^10*b^6 - 20*a^8*b^8 + 8*a^6*b^10 + a^4*b^12 - a^2*b^14 - 6*a^14*c^2 + 26*a^12*b^2*c^2 - 31*a^10*b^4*c^2 - a^8*b^6*c^2 + 20*a^6*b^8*c^2 - 8*a^4*b^10*c^2 + a^2*b^12*c^2 - b^14*c^2 + a^12*c^4 - 31*a^10*b^2*c^4 + 60*a^8*b^4*c^4 - 30*a^6*b^6*c^4 - 9*a^4*b^8*c^4 + 3*a^2*b^10*c^4 + 6*b^12*c^4 + 15*a^10*c^6 - a^8*b^2*c^6 - 30*a^6*b^4*c^6 + 32*a^4*b^6*c^6 - 3*a^2*b^8*c^6 - 15*b^10*c^6 - 20*a^8*c^8 + 20*a^6*b^2*c^8 - 9*a^4*b^4*c^8 - 3*a^2*b^6*c^8 + 20*b^8*c^8 + 8*a^6*c^10 - 8*a^4*b^2*c^10 + 3*a^2*b^4*c^10 - 15*b^6*c^10 + a^4*c^12 + a^2*b^2*c^12 + 6*b^4*c^12 - a^2*c^14 - b^2*c^14 : :

X(14934) lies on the cubic K725 and these lines: {3, 523}, {5, 14385}, {30, 110}, {74, 14480}, {265, 3154}, {1511, 7471}, {2072, 6760}, {5663, 14611}, {6070, 6699}, {14094, 14508}

X(14934) = midpoint of X(i) and X(j) for these {i,j}: {74, 14480}, {110, 477}, {14094, 14508}
X(14934) = reflection of X(i) in X(j) for these {i,j}: {265, 3154}, {6070, 6699}, {7471, 1511}

X(14935) =  X(11)X(4885)∩X(55)X(7123)

Barycentrics    a^2 (a-b-c) (b-c)^2 (a^2-2 a b+b^2+c^2) (a^2+b^2-2 a c+c^2) : :

X(14935) lies on the cubic K925 and these lines: {11,4885}, {55,7123}, {497,4554}, {1041,1937}, {1357,2821}, {1830,12723}, {3024,5581}, {3056,9432}, {3248,3270}

X(14935) = X(8817)-Ceva conjugate of X(650)
X(14935) = cevapoint of X(3022) and X(3271)
X(14935) = crosssum of X(i) and X(j) for these (i,j): {497, 3732}, {1633, 4000}
X(14935) = X(i)-isoconjugate of X(j) for these (i,j): {59, 3673}, {497, 7045}, {614, 4998}, {651, 3732}, {664, 1633}, {765, 7195}, {1275, 2082}, {4000, 4564}
X(14935) = barycentric product X(i)*X(j) for these {i,j}: {11, 7123}, {1037, 1146}, {2310, 7131}, {4858, 7084}
X(14935) = barycentric quotient X(i)/X(j) for these {i,j}: {663, 3732}, {1015, 7195}, {1037, 1275}, {2170, 3673}, {3022, 6554}, {3063, 1633}, {3271, 4000}, {7084, 4564}, {7123, 4998}

X(14936) =  ISOGONAL CONJUGATE OF X(1275)

Barycentrics    a^2 (b-c)^2 (-a+b+c)^2 : :

X(14936) lies on the Brocard inellipse and these lines: {1,8285}, {6,109}, {9,9365}, {11,650}, {19,800}, {32,607}, {39,2082}, {41,1017}, {42,1200}, {55,2195}, {100,294}, {115,5521}, {122,5517}, {187,1951}, {244,665}, {570,7300}, {577,2164}, {647,2611}, {649,3937}, {893,3512}, {910,8608}, {1146,2968}, {1197,9419}, {1210,1939}, {1407,11051}, {2092,2264}, {2114,2999}, {2207,7154}, {2241,7124}, {2310,3119}, {3002,5011}, {3003,7297}, {3035,5701}, {3124,5075}, {3271,6139}, {4534,11998}, {5540,13006}, {9367,12053}

X(14936) = isogonal conjugate of X(1275)
X(14936) = isogonal conjugate of the isotomic conjugate of of X(1146)
X(14936) = X(1021)-beth conjugate of X(11)
X(14936) = X(i)-complementary conjugate of X(j) for these (i,j): {7084, 513}, {7123, 3835}
X(14936) = X(i)-Ceva conjugate of X(j) for these (i,j): {6, 663}, {19, 512}, {41, 3709}, {55, 8641}, {57, 2488}, {200, 4524}, {220, 657}, {278, 2520}, {279, 513}, {294, 926}, {346, 3900}, {607, 3063}, {909, 1960}, {1146, 3270}, {1253, 10581}, {1436, 667}, {2164, 1946}, {2170, 3271}, {2258, 8653}, {2291, 6139}, {2310, 3022}, {4000, 6004}, {11051, 649}
X(14936) = bicentric difference of PU(103)
X(14936) = PU(103)-harmonic conjugate of X(6139)
X(14936) = barycentric square of X(650)
X(14936) = X(i)-isoconjugate of X(j) for these (i,j): {1, 1275}, {2, 7045}, {7, 4564}, {57, 4998}, {59, 85}, {65, 4620}, {69, 7128}, {75, 1262}, {99, 1020}, {100, 658}, {101, 4569}, {109, 4554}, {190, 934}, {269, 1016}, {279, 765}, {312, 7339}, {348, 7012}, {644, 4626}, {646, 6614}, {651, 664}, {653, 6516}, {662, 4566}, {668, 1461}, {693, 4619}, {738, 4076}, {927, 1025}, {1018, 4616}, {1042, 4601}, {1088, 1252}, {1254, 4590}, {1331, 13149}, {1407, 7035}, {1414, 4552}, {1415, 4572}, {1427, 4600}, {1446, 4570}, {2149, 6063}, {2171, 7340}, {3668, 4567}, {3699, 4617}, {3732, 8269}, {3952, 4637}, {4551, 4573}, {4557, 4635}, {4559, 4625}, {6064, 7147}, {7115, 7182}
X(14936) = X(294)-line conjugate of X(100)
X(14936) = crosspoint of X(i) and X(j) for these (i,j): {6, 663}, {25, 649}, {55, 650}, {200, 1021}, {220, 657}, {279, 513}, {346, 3900}, {2170, 2310}, {6591, 7151}
X(14936) = crossdifference of every pair of points on line {100, 658}
X(14936) = crosssum of X(i) and X(j) for these (i,j): {2, 664}, {6, 1633}, {7, 651}, {69, 190}, {100, 220}, {269, 1020}, {279, 658}, {394, 6516}, {934, 1407}, {4564, 7045}, {4566, 6354}, {6645, 6649}
X(14936) = barycentric product X(i)*X(j) for these {i,j}: {1, 2310}, {4, 3270}, {6, 1146}, {7, 3022}, {8, 3271}, {9, 2170}, {11, 55}, {21, 4516}, {25, 2968}, {33, 7004}, {41, 4858}, {56, 4081}, {57, 3119}, {59, 5532}, {60, 4092}, {115, 7054}, {158, 2638}, {200, 244}, {219, 8735}, {220, 1086}, {281, 7117}, {341, 3248}, {346, 1015}, {480, 1358}, {512, 7253}, {513, 3900}, {514, 657}, {522, 663}, {649, 3239}, {650, 650}, {652, 3064}, {661, 1021}, {667, 4397}, {693, 8641}, {764, 4578}, {885, 926}, {1019, 4171}, {1022, 14427}, {1043, 3122}, {1090, 1110}, {1098, 2643}, {1111, 1253}, {1260, 2969}, {1318, 4542}, {1357, 5423}, {1364, 1857}, {1365, 6061}, {1436, 5514}, {1565, 7071}, {2164, 6506}, {2287, 3125}, {2316, 4530}, {2326, 3708}, {2328, 3120}, {2332, 4466}, {3063, 4391}, {3124, 7058}, {3445, 4953}, {3572, 4148}, {3669, 4130}, {3676, 4105}, {3700, 7252}, {3709, 4560}, {3737, 4041}, {3937, 7046}, {3942, 7079}, {4124, 7077}, {4524, 7192}, {6065, 7336}, {6729, 6730}, {7151, 7358}, {7256, 8034}, {10501, 10502}, {11051, 13609}
X(14936) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 1275}, {11, 6063}, {31, 7045}, {32, 1262}, {41, 4564}, {55, 4998}, {60, 7340}, {200, 7035}, {220, 1016}, {244, 1088}, {284, 4620}, {480, 4076}, {512, 4566}, {513, 4569}, {522, 4572}, {649, 658}, {650, 4554}, {657, 190}, {663, 664}, {667, 934}, {798, 1020}, {884, 927}, {926, 883}, {1015, 279}, {1019, 4635}, {1021, 799}, {1146, 76}, {1253, 765}, {1356, 7143}, {1357, 479}, {1364, 7055}, {1397, 7339}, {1919, 1461}, {1946, 6516}, {1973, 7128}, {1977, 1407}, {2170, 85}, {2175, 59}, {2212, 7012}, {2287, 4601}, {2310, 75}, {2328, 4600}, {2638, 326}, {2968, 305}, {3022, 8}, {3063, 651}, {3119, 312}, {3121, 1427}, {3122, 3668}, {3124, 6354}, {3125, 1446}, {3239, 1978}, {3248, 269}, {3270, 69}, {3271, 7}, {3709, 4552}, {3733, 4616}, {3737, 4625}, {3900, 668}, {3937, 7056}, {4079, 4605}, {4081, 3596}, {4105, 3699}, {4130, 646}, {4171, 4033}, {4397, 6386}, {4459, 7205}, {4516, 1441}, {4524, 3952}, {6061, 6064}, {6591, 13149}, {7004, 7182}, {7054, 4590}, {7063, 181}, {7117, 348}, {7252, 4573}, {7253, 670}, {8638, 2283}, {8641, 100}, {8735, 331}, {9447, 2149}, {14827, 1252}

X(14937) =  X(5)X(76)∩X(670)X(10010)

Barycentrics    b^2*c^2*(2*a^8*b^4 - 2*a^6*b^6 + 2*a^8*b^2*c^2 + 5*a^6*b^4*c^2 - a^2*b^8*c^2 + 2*a^8*c^4 + 5*a^6*b^2*c^4 + 15*a^4*b^4*c^4 + a^2*b^6*c^4 + b^8*c^4 - 2*a^6*c^6 + a^2*b^4*c^6 - 2*b^6*c^6 - a^2*b^2*c^8 + b^4*c^8) : :

X(14937) lies on these lines: {5, 76}, {670, 10010}, {3734, 14382}

X(14937) = {X(76),X(7697)}-harmonic conjugate of X(327)

X(14938) =  ISOGONAL CONJUGATE OF X(1199)

Barycentrics    (SB^2-4*R^2*SB-3*S^2)*(SC^2-4* R^2*SC-3*S^2) : :

See Antreas Hatzipolakis and César Lozada, Hyacinthos 26705.

X(14938) lies on these lines: {5, 97}, {233, 14627}, {394, 1656}, {1073, 5070}, {3926, 14786}

X(14938) = isogonal conjugate of X(1199)

X(14939) =  (name pending)

Barycentrics    (a^2-b^2-c^2) (4 a^14-7 a^12 b^2-8 a^10 b^4+21 a^8 b^6-4 a^6 b^8-13 a^4 b^10+8 a^2 b^12-b^14-7 a^12 c^2+34 a^10 b^2 c^2-25 a^8 b^4 c^2-52 a^6 b^6 c^2+71 a^4 b^8 c^2-14 a^2 b^10 c^2-7 b^12 c^2-8 a^10 c^4-25 a^8 b^2 c^4+112 a^6 b^4 c^4-58 a^4 b^6 c^4-48 a^2 b^8 c^4+27 b^10 c^4+21 a^8 c^6-52 a^6 b^2 c^6-58 a^4 b^4 c^6+108 a^2 b^6 c^6-19 b^8 c^6-4 a^6 c^8+71 a^4 b^2 c^8-48 a^2 b^4 c^8-19 b^6 c^8-13 a^4 c^10-14 a^2 b^2 c^10+27 b^4 c^10+8 a^2 c^12-7 b^2 c^12-c^14) : :

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 26709.

X(14939) lies on this line: {20,64}

X(14940) =  18th HATZIPOLAKIS-MOSES-EULER POINT

Barycentrics    (a^2+b^2-c^2) (a^2-b^2+c^2) (a^6-3 a^4 b^2+3 a^2 b^4-b^6-3 a^4 c^2+a^2 b^2 c^2+b^4 c^2+3 a^2 c^4+b^2 c^4-c^6) : :

Let La be the polar of X(4) wrt the circle centered at A and passing through X(5). Define Lb and Lc cyclically. (Note: X(4) is the perspector of any circle centered at a vertex of ABC.) Let A' = Lb∩Lc, and define B' and C' cyclically. Triangle A'B'C' is homothetic to ABC, with center of homothety X(14940). (Randy Hutson, November 2, 2017)

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 26711.

X(14940) lies on these lines: {2,3}, {6,13418}, {49,11264}, {110,5449}, {112,7749}, {113, 11440}, {125,1614}, {156,3448}, {252,933}, {562,12044}, {1199, 13567}, {1506,10312}, {1986, 11591}, {2888,11597}, {3043, 5972}, {3054,8744}, {3580,9820}, {5876,7722}, {5944,11565}, {6152,13365}, {6188,14249}, {6247,12112}, {9927,11449}, {10104,14675}, {10182,14644}, {10540,13561}, {10576,10881}, {10577,10880}, {11202,12289}, {11447,13970}, {11448,13909}, {11464,11704}, {11468,12244}, {11562,12111}, {13399,14862}, {13450,14165}

X(14940) = orthocentroidal-circle-inverse of X(6143)
X(14940) = X(93)-Ceva conjugate of X(4)
X(14940) = polar conjugate of X(13585)
X(14940) = barycentric quotient X(4)/X(13585)
X(14940) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 4, 6143), (2, 20, 6640), (2, 3549, 631), (2, 7505, 4), (2, 7558, 3525), (5, 186, 4), (5, 10018, 186), (5, 10125, 3), (24, 1656, 7577), (24, 7577, 4), (140, 403, 3520), (235, 14865, 4), (403, 3520, 4), (468, 1594, 3518), (468, 3628, 1594), (631, 3090, 6816), (858, 13383, 12088), (1594, 3518, 4), (1658, 10255, 3153), (2072, 10020, 7488), (3090, 3147, 4), (3515, 5055, 7547), (6640, 10201, 20), (9927, 11449, 12383), (14784, 14785, 10255)

leftri

Brocard-Lemoine points: X(14941)-X(14952)

rightri

This preamble and centers X(14941)-X(14952) were contributed by César Eliud Lozada, October 24-27, 2017.

Let ABC be a triangle, P any point in its plane (not on its sidelines) and A', B', C' the traces of P on ABC. Let Ab, Ac be the points where BC is cut by the parallel to AB through B' and the parallel to AC through C', respectively. Build Bc, Ba, Ca, Cb cyclically. Then lines AAb, BBc, CCa concur at a point W1 and lines AAc, BBa, CCb concur at a point W2.

For P = u : v : w (trilinears), W1 and W2 have trilinears as shown:
 W1 = c/v : a/w : b/u
 W2 = b/w : c/u : a/v

Note that for P=X(6), W1 and W2 are the Brocard points of ABC.

The previous construction is due to Lemoine as an attempt to generalize the Brocard geometry (Roger A. Johnson, Advanced Eucliden Geometry, Dover, New York, 1960, §499-500, pp. 299). Johnson writes, these points have many properties resembling those of the Brocard points.

Some properties:

 a) For P=X(2), W1=W2=X(2)

 b) Points Ab, Ac, Bc, Ba, Ca, Cb lie on a conic (here named the Brocard-Lemoine conic of P) with center O(P) given by:
   O(P) = (a^4*u^4-a*u*(b*v+c*w)*(3*b*c*v*w+a^2*u^2)-(b^2*v^2+c^2*w^2)*(b*c*v*w+3*a^2*u^2)-b*v*c*w*(-b*c*v*w+2*a^2*u^2))/a : : (trilinears)
   O(P) = x^4-x*(y+z)*(x^2+3*y*z)-(y^2+z^2)*(3*x^2+y*z)-y*z*(2*x^2-y*z) : : (barycentrics)

     - For P=X(194), the conic is a circle centered at X(14135) and having squared-radius = ∏(3*S^2-4*SB*SC-SW^2)/(64*R^2*S^4)

 c) The trilinear pole Q(P) of the line W1W2 has coordinates:
   Q(P) = u*(b^2*v^2-a*c*u*w)*(c^2*w^2-a*b*u*v) : : (trilinears)
   Q(P) = x*(y^2-x*z)*(z^2-x*y) : : (barycentrics for P=x:y:z)

     - For P on the line at infinity, Q(P) = P
     - For P on the Steiner circumellipse, Q(P) = X(2)
     - For any other P, Q(Q(P)) = P, i.e., pairs {P, Q(P)} form a conjugation.

  Following the last property, Q(P) will be named here the antitomic conjugate of P. This conjugacy was mentioned by Randy Hutson in X(9513), although its explanation was left pending. In a personal communication (Oct 26, 2017), Hutson gives the reasons for this name:

As the antigonal conjugate of a point P is defined as:
 1) The isogonal conjugate of the inverse in the circumcircle (=isogonal conjugate of the line-at-infinity) of the isogonal conjugate of P,

I define the antitomic conjugate of P as:
 1) The isotomic conjugate of the inverse in the Steiner-circumellipse (=isotomic conjugate of the line-at-infinity) of the isotomic conjugate of P.

  Randy Hutson also notes that Q(P) is the antipode of P in the hyperbola {A,B,C,X(2),P}.

The appearance of {I,J} in the following list means that Q(X(I)) = X(J) (for P not at infinity and not on the Steiner circumellipse):
{1, 291}, {3, 14941}, {4, 98}, {6, 694}, {7, 673}, {8, 14942}, {9, 14943}, {10, 11599}, {13, 14}, {14, 13}, {17, 11602}, {18, 11603}, {20, 14944}, {31, 14945}, {32, 14946}, {69, 287}, {75, 335}, {76, 1916}, {83, 11606}, {86, 6650}, {92, 1952}, {98, 4}, {100, 14947}, {110, 9513}, {111, 14948}, {226, 11608}, {239, 350}, {253, 6330}, {256, 7168}, {264, 1972}, {287, 69}, {291, 1}, {297, 325}, {298, 11092}, {299, 11078}, {321, 11611}, {325, 297}, {335, 75}, {350, 239}, {385, 3978}, {673, 7}, {694, 6}, {1016, 6630}, {1031, 9483}, {1432, 7167}, {1916, 76}, {1944, 1948}, {1948, 1944}, {1952, 92}, {1972, 264}, {1987, 1988}, {1988, 1987}, {2319, 4876}, {2394, 14223}, {2994, 8777}, {2996, 8781}, {3505, 6660}, {3912, 9436}, {3978, 385}, {4440, 6634}, {4876, 2319}, {5466, 9180}, {5485, 5503}, {5503, 5485}, {6330, 253}, {6630, 1016}, {6634, 4440}, {6650, 86}, {6660, 3505}, {7167, 1432}, {7168, 256}, {8777, 2994}, {8781, 2996}, {9178, 14606}, {9180, 5466}, {9302, 14492}, {9436, 3912}, {9473, 9476}, {9476, 9473}, {9483, 1031}, {9513, 110}, {11599, 10}, {11606, 83}, {11608, 226}, {11611, 321}, {14223, 2394}, {14492, 9302}, {14606, 9178}, {14941, 3}, {14942, 8}, {14943, 9}, {14944, 20}, {14945, 31}, {14946, 32}, {14947, 100}, {14948, 111}

The appearance of (I,J) in the following list means that O(X(I)) = X(J): (1,14949), 6,14950), (8,14951), (20,14952), (194,14135)

The antitomic conjugate of the circumcircle is the isogonal conjugate of the Brocard circle. (Randy Hutson, January 17, 2020)

X(14941) = ANTITOMIC CONJUGATE OF X(3)

Trilinears    (cos A)/(sin^2 2A - sin 2B sin 2C) : :
Barycentrics    (4*SW*R^2-SW^2+S^2-2*SA*SB)*(4*SW*R^2-SW^2+S^2-2*SA*SC)*SA*(SB+SC) : :

X(14941) lies on the Johnson circumconic, cubics K357, K779 and these lines: {2,1972}, {3,1625}, {5,276}, {97,110}, {394,6638}, {1073,1351}, {1214,1956}, {2080,6760}, {2797,6321}, {3095,14059}

X(14941) = reflection of X(6528) in X(5)
X(14941) = antipode of X(6528) in Johnson circumconic
X(14941) = trilinear pole of the line {216, 520}
X(14941) = cevapoint of: (418, 3289), (684, 2972)
X(14941) = X(511)-crossconjugate of X(3)
X(14941) = X(6528)-of-Johnson-triangle
X(14941) = antipode of X(3) in hyperbola {A,B,C,X(2),X(3)}
X(14941) = X(i)-isoconjugate-of-X(j) for these {i,j}: {4, 1955}, {19, 401}, {92, 1971}, {275, 2313}

X(14942) = ANTITOMIC CONJUGATE OF X(8)

Barycentrics    (a^2-c*a+b*(b-c))*(a^2-b*a-c*(b-c))*(-a+b+c) : :

X(14942) lies on the cubics K296 and K770, on the circumconic with center X(1), and on these lines: {1,85}, {2,11}, {8,220}, {10,10482}, {20,963}, {29,1855}, {33,92}, {48,5236}, {56,9305}, {64,962}, {75,4319}, {80,5377}, {86,2293}, {103,516}, {145,10405}, {171,1416}, {189,1814}, {190,2310}, {200,312}, {257,3057}, {333,643}, {514,10695}, {517,1952}, {518,10025}, {519,666}, {740,7281}, {884,8851}, {885,1320}, {894,14100}, {918,1280}, {919,1311}, {940,1462}, {950,1220}, {1010,4314}, {1027,1120}, {1043,7257}, {1088,2263}, {1936,2342}, {2319,6169}, {2785,7983}, {2809,3732}, {2994,7072}, {3022,14839}, {3684,5853}, {3685,3693}, {3729,4907}, {3750,13405}, {3891,9539}, {3912,6185}, {3957,6742}, {4307,14548}, {4326,10436}, {4336,4360}, {4645,9453}, {4997,6745}, {7169,10431}

X(14942) = reflection of X(i) in X(j) for these (i,j): (8, 1146), (664, 1)
X(14942) = isogonal conjugate of X(1458)
X(14942) = isotomic conjugate of X(9436)
X(14942) = polar conjugate of X(5236)
X(14942) = trilinear pole of the line {9, 522}
X(14942) = X(1043)-beth conjugate of X(644)
X(14942) = Cevapoint of (i,j) for these (i,j): (1, 516), (8, 3685), (9, 2340)
X(14942) = X=X(294)-cross conjugate of X(673)
X(14942) = X(i)-isoconjugate-of-X(j) for these (i, j): {1, 1458}, {3, 1876}, {6, 241}, {7, 2223}, {34, 1818}, {48,5236}, {56, 518}, {57, 672}, {59, 3675}, {65, 3286}, {77, 2356}, {105, 1362}, {109, 2254}, {162, 6130}, {269, 2340}, {513, 2283}, {603, 1861}, {604, 3912}, {649, 1025}, {651, 665}, {667, 883}, {918, 1415}, {926, 934}, {1106, 3717}, {1397, 3263}, {1407, 3693}, {1408, 3932}, {1412, 3930}, {1416, 4712}, {1429, 3252}, {1431, 4447} et als.
X(14942) = X(1)-zayin conjugate of X(1458)
X(14942) = X(105)-anticomplementary conjugate of X(152)
X(14942) = inverse-in-Feuerbach-hyperbola of X(673)
X(14942) = excentral-to-ABC barycentric image of X(1282)
X(14942) = {X(105), X(13576)}-harmonic conjugate of X(673)

X(14943) = ANTITOMIC CONJUGATE OF X(9)

Barycentrics    (b*a^3-(2*b^2-c^2)*a^2+(b-c)*(b^2+b*c+2*c^2)*a+c^2*(b-c)^2)*(c*a^3+(b^2-2*c^2)*a^2-(b-c)*(2*b^2+b*c+c^2)*a+b^2*(b-c)^2)*(-a+b+c)*a : :

X(14943) lies on these lines: {2,3119}, {9,1742}, {100,6605}, {142,4569}

X(14943) = reflection of X(4569) in X(142)
X(14943) = trilinear pole of the line {1212, 3900}
X(14943) = X(518)-crossconjugate of X(9)
X(14943) = X(i)-isoconjugate of X(j) for these {i,j}: {6,14189}, {56,10025}, {57,9441}

X(14944) = ANTITOMIC CONJUGATE OF X(20)

Barycentrics    SB*SC*(S^2-2*SB*SC)*((SA+SB)*S^2-2*SA*SB*SW)*((SC+SA)*S^2-2*SW*SC*SA) : :

X(14944) lies on the cubic K718 and these lines: {2,107}, {325,1529}, {5895,9289}

X(14944) = trilinear pole of the line {1249, 8057}
X(14944) = X(i)-isoconjugate-of-X(j) for these {i, j}: {64,8766}, {441, 2155}, {1073, 2312}

X(14945) = ANTITOMIC CONJUGATE OF X(31)

Barycentrics    (c*a-b^2)*(c^2*a^2+b^2*c*a+b^4)*(b*a-c^2)*(b^2*a^2+b*c^2*a+c^4)*a^3 : :

X(14945) lies on these lines: {723,815}, {743,795}

X(14945) = X(794)-isoconjugate of X(4586)

X(14946) = ANTITOMIC CONJUGATE OF X(32)

Barycentrics    a^4*(b^8-a^4*c^4)*(c^8-a^4*b^4) : :

X(14946) lies on the cubic K354 and these lines: {39,695}, {711,805}, {733,783}, {1916,8783}, {4630,8023}

X(14946) = anticomplement of X(39082)
X(14946) = trilinear pole of the line {688, 8265}
X(14946) = X(i)-isoconjugate-of-X(j) for these {i, j}: {384, 1966}, {385, 1965}, {782, 4593}, {1580, 9230}, {1582, 3978}, {1691, 1925}, {1915, 1926}, {1932, 14603}

X(14946) = center of bianticevian conic of PU(1)

X(14947) = ANTITOMIC CONJUGATE OF X(100)

Barycentrics    (b*a^3-2*b^2*a^2+(b^3+b*c^2-c^3)*a-c^3*(b-c))*(c*a^3-2*c^2*a^2-(b^3-b^2*c-c^3)*a+b^3*(b-c))*a : :

X(14947) lies on the Feuerbach hyperbola, cubic K359 and these lines: {1,1025}, {2,885}, {7,3675}, {9,1026}, {11,2481}, {21,1083}, {55,5377}, {100,294}, {104,2808}, {1172,4238}, {1320,14839}, {2414,6601}, {2795,10769}

X(14947) = reflection of X(i) in X(j) for these (i,j): (100, 6184), (2481, 11)
X(14947) = antigonal conjugate of X(2481)
X(14947) = isogonal conjugate of X(5091)
X(14947) = antipode of X(2481) in Feuerbach hyperbola
X(14947) = X(i)-isoconjugate-of-X(j) for these {i, j}: {1, 5091}, {6, 9318}
X(14947) = trilinear pole of the line {518, 650}

X(14948) = ANTITOMIC CONJUGATE OF X(111)

Barycentrics    (b^2*a^4+(b^4-5*b^2*c^2+2*c^4)*a^2+c^4*(2*b^2-c^2))*(c^2*a^4+(2*b^4-5*b^2*c^2+c^4)*a^2-b^4*(b^2-2*c^2))*a^2 : :

X(14948) lies on the cubic K688 and these lines: {6,11634}, {111,1084}, {126,670}, {543,3228}, {694,2854}, {888,9156}, {6088,14606}, {9178,14263}

X(14948) = reflection of X(i) in X(j) for these (i,j): (111, 1084), (670, 126)
X(14948) = antigonal conjugate of X(670)
X(14948) = isogonal conjugate of X(5108)
X(14948) = X(1)-isoconjugate of X(5108)
X(14948) = trilinear pole of the line {512, 3291}

X(14949) = CENTER OF THE BROCARD-LEMOINE CONIC OF X(1)

Barycentrics    a^4-(b+c)*a^3-(3*b^2+2*b*c+3*c^2)*a^2-3*b*c*(b+c)*a-b*c*(b^2-b*c+c^2) : :

This conic is the bicevian conic of PU(6).

X(14949) lies on these lines: {1,6646}, {226,6625}, {239,257}

X(14950) = CENTER OF THE BROCARD-LEMOINE CONIC OF X(6)

Barycentrics    a^8-(b^2+c^2)*a^6-(3*b^4+2*b^2*c^2+3*c^4)*a^4-3*b^2*c^2*(b^2+c^2)*a^2-b^2*c^2*(b^4-b^2*c^2+c^4) : :

This conic is the bicevian conic of PU(1).

X(14950) lies on these lines: {6,6655}, {385,1194}, {427,1031}

X(14951) = CENTER OF THE BROCARD-LEMOINE CONIC OF X(8)

Barycentrics    (b^2-4*b*c+c^2)*a^2-(b+c)*(b^2-5*b*c+c^2)*a+b*c*(b^2-4*b*c+c^2) : :

This conic is the bicevian conic of the isogonal conjugates of PU(48).

X(14951) lies on these lines: {8,726}, {10,7187}, {85,10009}, {330,4384}, {3061,3452}

X(14952) = CENTER OF THE BROCARD-LEMOINE CONIC OF X(20)

Barycentrics    (-a^2+b^2+c^2)*(5*a^8-5*(b^2+c^2)*a^6-(13*b^4-30*b^2*c^2+13*c^4)*a^4+13*(b^4-c^4)*(b^2-c^2)*a^2-12*(b^2-c^2)^2*b^2*c^2) : :

X(14952) lies on these lines: {20,3564}, {441,6337}

leftri

Bicentrically induced harmonic conjugates: X(14953)-X(14966)

rightri

This preamble and centers X(14953)-X(14966) were contributed by Clark Kimberling and Peter Moses, October 25, 2017.

Using barycentric coordinates, suppose that P = p : q : r and U = u : v : w are a bicentric pair of points. Let

P' = line p*x + q*y + r*z = 0
U' = line u*x + v*y + w*z = 0
L = a line
P* = L∩P'
U* = L∩U'
X = a point on L
X' = (P*,U*)-harmonic conjugate of X, to be expressed as L(P,U)-harmonic conjugate of X

If L is a central line and X is a triangle center, then X' is a triangle center. For example, if X is on the Euler line, then X' is on the Euler line, for every choice of bicentric pair (P,U), and the point X' in this case is written as Euler(P,U)-harmonic conjugate of X. If X is on the Brocard axis, then X' is written as Brocard(P,U)-harmonic conjugate of X.


X(14953) = EULER(P(6),U(6))-HARMONIC CONJUGATE OF X(2)

Barycentrics    (a+b) (a+c) (2 a^3-a^2 b-b^3-a^2 c+b^2 c+b c^2-c^3) : :
X(14953) = X(2) - 4 X(1375)

If you have Geogebra, you can view X(14953).

X(14953) lies on these lines: {2,3}, {7,284}, {57,3188}, {58,5222}, {81,279}, {110,2724}, {144,2287}, {239,514}, {333,5792}, {347,1172}, {534,3007}, {673,3286}, {962,2360}, {1086,3285}, {1333,4000}, {1448,5256}, {1474,4329}, {1790,8025}, {2173,8680}, {2194,3474}, {2303,3672}, {2328,9778}, {4273,4644}, {4384,4652}, {4653,5308}, {6360,9536}

X(14953) = midpoint of X(5196) and X(7479)
X(14953) = reflection of X(i) in X(j) for these {i,j}: {857, 1375}, {14543, 2173}
X(14953) = anticomplement X(857)
X(14953) = cevapoint of X(516) and X(910)
X(14953) = crossdifference of every pair of points on line {42, 647}
X(14953) = X(1434)-daleth conjugate of X(81)
X(14953) = X(i)-isoconjugate of X(j) for these (i,j): {10, 911}, {37, 103}, {65, 2338}, {661, 677}, {1018, 2424}, {1815, 1824}
X(14953) = barycentric product X(i)X(j) for these {i,j}: {86, 516}, {99, 676}, {274, 910}, {314, 1456}, {2398, 7192}, {4025, 4241}
X(14953) = barycentric quotient X(i)/X(j) for these {i,j}: {58, 103}, {110, 677}, {284, 2338}, {516, 10}, {676, 523}, {910, 37}, {1333, 911}, {1456, 65}, {1790